MCThermal

Computes the thermal conductivity and specific heat for (U,Pu)C (mixed mono-carbide) fuels based on mole fractions, porosity, and temperature.

Description

The material model MCThermal calculates the thermal conductivity $var element = document.getElementById("moose-equation-e0c4e027-0b67-4c9f-b830-f7386cb73087");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ and specific heat capacity $var element = document.getElementById("moose-equation-37915688-1767-47ff-9950-2cb9ceaca05d");katex.render("c_p", element, {displayMode:false,throwOnError:false});$ of U $var element = document.getElementById("moose-equation-f3a1ddbb-fe7a-4376-a7c4-e4ceae926f14");katex.render("_{1-x}", element, {displayMode:false,throwOnError:false});$ Pu $var element = document.getElementById("moose-equation-45afb0d5-47ed-46b9-aa52-30b9486932a7");katex.render("_x", element, {displayMode:false,throwOnError:false});$ C, where $var element = document.getElementById("moose-equation-dd1658c2-e490-4847-9c6f-f78d1fe6ffd6");katex.render("x", element, {displayMode:false,throwOnError:false});$ refers to the atomic ratio of Pu to U. Input is given as $var element = document.getElementById("moose-equation-de8f33c5-4d1e-4066-aad0-46137a601977");katex.render("X_{Pu}", element, {displayMode:false,throwOnError:false});$ , which is the mole fraction of Pu and is therefore equivalent to $var element = document.getElementById("moose-equation-f67b1211-ddc7-41e8-a2ba-00699b233f87");katex.render("x", element, {displayMode:false,throwOnError:false});$ .

Two thermal conductivity models are provided, Matzke (Matzke, 1986) and Steiner (Steiner, 1975), while only one specific heat model is provided, Preusser (Preusser, 1982).

Matzke Thermal Conductivity

The thermal conductivity $var element = document.getElementById("moose-equation-cf870db1-c045-4303-97a0-2953a8c09984");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ for UC and (U $var element = document.getElementById("moose-equation-ab7dab23-36fd-4c05-9be1-ef228084a237");katex.render("_{0.8}", element, {displayMode:false,throwOnError:false});$ ,Pu $var element = document.getElementById("moose-equation-1b12585f-5bcc-478e-bbc0-7bf6f519f252");katex.render("_{0.2}", element, {displayMode:false,throwOnError:false});$ )C is provided in Matzke (1986) as $var element = document.getElementById("moose-equation-e6b962d6-8c05-4eb0-9ab1-03094d0b4075");katex.render("\\begin{aligned} \\lambda_{UC} &= \\begin{cases} 21.7 - 3.04\\times 10^{-3} (T-273.15) + 3.61\\times 10^{-6} (T-273.15)^2, & 323 < T < 973 \\text{\\ K}; \\\\ 20.2 + 1.48\\times 10^{-3} (T-273.15), & 973 < T < 2573 \\text{\\ K}; \\\\ \\end{cases} \\\\ \\lambda_{(U_{0.8},Pu_{0.2})C} &= \\begin{cases} 17.5 - 5.65\\times 10^{-3} (T-273.15) + 8.14\\times 10^{-6} (T-273.15)^2, & 323 < T < 773 \\text{\\ K}; \\\\ 12.76 + 8.71\\times 10^{-3} (T-273.15) - 1.88\\times 10^{-6} (T-273.15)^2, & 773 < T < 2573 \\text{\\ K}. \\\\ \\end{cases} \\end{aligned}", element, {displayMode:true,throwOnError:false});$ Here, $var element = document.getElementById("moose-equation-a527bd69-7409-409f-a225-9438aae45fbe");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ is provided in (W/m/K) and $var element = document.getElementById("moose-equation-156fd333-a38c-4912-8158-5ba9259c1e63");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature in (K). The sharp transition is smoothed using the SmootherStep function from 948 K to 990 K for $var element = document.getElementById("moose-equation-1ae0dbf5-2476-4638-a0cd-6f59242f7750");katex.render("\\lambda_{UC}", element, {displayMode:false,throwOnError:false});$ and 773 K to 798 K for $var element = document.getElementById("moose-equation-c96a550c-6216-402b-acd0-92c296616279");katex.render("\\lambda_{(U_{0.8},Pu_{0.2})C}", element, {displayMode:false,throwOnError:false});$ The thermal conductivities for the pure uranium carbide and mixed carbides are used to linearly extrapolate as a function of plutonium content by, $var element = document.getElementById("moose-equation-1c3b4512-1c1f-4d87-ada9-6fc2b378839b");katex.render(" \\frac{\\lambda_{(U_{0.8},Pu_{0.2})C} - \\lambda_{UC}}{0.2} + \\lambda_{UC}.", element, {displayMode:true,throwOnError:false});$ While rudimentary, and lacking data to enforce the simplified linear extrapolation, such a correlation is adequate for simple analysis at this point.

Steiner Thermal Conductivity

In addition to the Blank (2006) model, Steiner's model (Steiner, 1975) suggested by Preusser (1982) is also available: $var element = document.getElementById("moose-equation-2d013560-fb65-4996-84a4-178c9bc83d99");katex.render("\\begin{aligned} \\lambda_{\\text{UC}} = \\begin{cases} 20.0, & T < 773 \\text{\\ K}; \\\\ 20.0 + 1.3\\times 10^{-3} (T-773), & T > 773 \\text{\\ K}; \\\\ \\end{cases}\\\\ \\lambda_{(\\text{U}_{0.8},\\text{Pu}_{0.2})C} = \\begin{cases} 12.6, & T < 773 \\text{\\ K}; \\\\ 12.6 + 4.1\\times 10^{-3} (T-773), & T > 773 \\text{\\ K}. \\\\ \\end{cases}\\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false});$ Here, $var element = document.getElementById("moose-equation-a3a89608-439c-403c-a355-b5a997b9b46b");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ is provided in (W/m/K) and $var element = document.getElementById("moose-equation-bec9e409-bbac-4d73-af49-b25c26bf5ecf");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature in (K). The sharp transition is smoothed using the SmootherStep function from 773 K to 798 K. The thermal conductivities for the pure uranium carbide and mixed carbides are used to linearly extrapolate as a function of plutonium content by, $var element = document.getElementById("moose-equation-004c4fd1-f14d-458f-b216-5fb63e889cb4");katex.render(" \\frac{\\lambda_{(\\text{U}_{0.8},\\text{Pu}_{0.2})C} - \\lambda_{\\text{UC}}}{0.2} + \\lambda_{\\text{UC}}.", element, {displayMode:true,throwOnError:false});$ While rudimentary, and lacking data to enforce the simplified linear extrapolation, such a correlation is adequate for simple analysis at this point.

Thermal Conductivity Porosity Correction

A porosity correction is applied to the thermal conductivity following Blank (2006): $var element = document.getElementById("moose-equation-0d3891ad-ab0e-4131-bd51-e0b002f107b3");katex.render(" \\lambda(T, x, \\phi) = \\lambda_0(T, x) f(\\phi), \\\\ f(\\phi) = \\frac{1-\\phi}{1+\\phi}", element, {displayMode:true,throwOnError:false});$ Here, $var element = document.getElementById("moose-equation-97df02a2-800d-49cb-bcc0-de42a32c197b");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ is the thermal conductivity of the fuel as a function of temperature $var element = document.getElementById("moose-equation-1b76e61a-8b3d-4d68-af48-dd0292015aeb");katex.render("T", element, {displayMode:false,throwOnError:false});$ , composition $var element = document.getElementById("moose-equation-54a45002-52ea-4325-b9a4-f348230f3784");katex.render("x", element, {displayMode:false,throwOnError:false});$ , and fractional porosity, $var element = document.getElementById("moose-equation-9ef84a33-c988-4ada-a7eb-2678f48b2948");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$ , given the fresh-fuel thermal conductivity $var element = document.getElementById("moose-equation-415f1b40-a9d2-4710-884a-8593db024045");katex.render("\\lambda_0", element, {displayMode:false,throwOnError:false});$ , and the porosity correction $var element = document.getElementById("moose-equation-a9f44988-51f1-4f6d-87f0-5a734942ceec");katex.render("f", element, {displayMode:false,throwOnError:false});$ .

Specific Heat Capacity

Specific heat capacity $var element = document.getElementById("moose-equation-4c401538-da20-4af9-bd1e-0dd07cbfbfbf");katex.render("c_p", element, {displayMode:false,throwOnError:false});$ is calculated as a function of temperature and of plutonium fraction according to the heat capacities of UC and (U $var element = document.getElementById("moose-equation-ce45d420-4f62-4e99-a05d-95bb35804dbd");katex.render("_{0.8}", element, {displayMode:false,throwOnError:false});$ ,Pu $var element = document.getElementById("moose-equation-b64e4266-af42-4b8e-958b-ed7b0ae88a54");katex.render("_{0.2}", element, {displayMode:false,throwOnError:false});$ )C suggested by Preusser (1982), and repeated in Matzke (1986): $var element = document.getElementById("moose-equation-aaf1d43c-4770-4f2a-bdbe-0f514f2c70fc");katex.render("\\begin{aligned} C_{p,\\text{UC}} = 54.4 + 9.62\\times 10^{-3} T, \\\\ C_{p,(\\text{U}_{0.8},\\text{Pu}_{0.2})\\text{C}} = 50.2 + 9.62\\times 10^{-3} T. \\end{aligned}", element, {displayMode:true,throwOnError:false});$ Here, $var element = document.getElementById("moose-equation-d600a8bf-15b6-441b-8092-d260cea3e312");katex.render("C_p", element, {displayMode:false,throwOnError:false});$ is provided in (J/mol/K) and $var element = document.getElementById("moose-equation-a85f2d6b-c612-4bb0-8dda-ed160ddd264b");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature in (K). To convert to specific heat capacity, $var element = document.getElementById("moose-equation-51459572-a515-43e8-955f-4b1bded16cce");katex.render("C_p", element, {displayMode:false,throwOnError:false});$ is divided by the atomic mass, $var element = document.getElementById("moose-equation-3a1e247e-f1a9-430d-b4f2-40973b5d6451");katex.render(" c_p = C_p / (A_U (1 - X_{Pu}) + A_{Pu} X_{Pu} + A_C),", element, {displayMode:true,throwOnError:false});$ where the elemental atomic masses ( $var element = document.getElementById("moose-equation-a425e746-f8a9-4f4b-8c73-891f197c329c");katex.render("A", element, {displayMode:false,throwOnError:false});$ ) are in (kg/mol).

The specific heats for the pure uranium carbide and mixed carbides are used to linearly extrapolate as a function of plutonium content by, $var element = document.getElementById("moose-equation-baa1822a-25d5-45c2-9b0e-db1e75d0c674");katex.render(" \\frac{c_{p,(\\text{U}_{0.8},\\text{Pu}_{0.2})\\text{C}} - c_{p,\\text{UC}}}{0.2} + c_{p,\\text{UC}}.", element, {displayMode:true,throwOnError:false});$ While rudimentary and lacking data to enforce the simplified linear extrapolation, such a correlation is adequate for simple analysis at this point.

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [thermal]
    type = MCThermal<<<{"description": "Computes the thermal conductivity and specific heat for (U,Pu)C (mixed mono-carbide) fuels based on mole fractions, porosity, and temperature.", "href": "MCThermal.html"}>>>
    temperature<<<{"description": "Coupled temperature."}>>> = temp
    X_Pu<<<{"description": "Mole fraction of Pu, which is the same as x in the common notation (U_{1-x},Pu_x)C."}>>> = 0.1
    porosity<<<{"description": "Porosity material property name."}>>> = 0.1
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
[]

Input Parameters

X_PuMole fraction of Pu, which is the same as x in the common notation (U_{1-x},Pu_x)C.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Mole fraction of Pu, which is the same as x in the common notation (U_{1-x},Pu_x)C.
porosityPorosity material property name.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Porosity material property name.
temperatureCoupled temperature.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled 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.
execute_onLINEARThe list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
Default:LINEAR
C++ Type:ExecFlagEnum
Options:XFEM_MARK, NONE, INITIAL, LINEAR, NONLINEAR_CONVERGENCE, NONLINEAR, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, FINAL, CUSTOM
Controllable:No
Description:The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
specific_heat_modelPREUSSERThe model to be used for thermal conductivity and specific heat.
Default:PREUSSER
C++ Type:MooseEnum
Options:PREUSSER
Controllable:No
Description:The model to be used for thermal conductivity and specific heat.
thermal_conductivity_modelMATZKEThe model to be used for thermal conductivity and specific heat.
Default:MATZKE
C++ Type:MooseEnum
Options:MATZKE, STEINER
Controllable:No
Description:The model to be used for thermal conductivity and specific heat.
value_range_behaviorEXCEPTIONWhat to do if input value is outside the range of applicability.
Default:EXCEPTION
C++ Type:MooseEnum
Options:ERROR, WARN, IGNORE, EXCEPTION
Controllable:No
Description:What to do if input value is outside the range of applicability.

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

(test/tests/mc_thermal/nonad.i)
(test/tests/mc_thermal/preusser_spheat_exact.i)
(test/tests/mc_thermal/thcond_porosity_exact.i)
(test/tests/mc_thermal/steiner_thcond_exact.i)
(test/tests/mc_thermal/matzke_thcond_exact.i)
(test/tests/mc_thermal/ad.i)
(assessment/carbide/EBRII/WSA32/analysis/base.i)

References

Hubert Blank. Nonoxide Ceramic Nuclear Fuels, chapter, pages. John Wiley & Sons, Ltd, 2006. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9783527603978.mst0108, doi:10.1002/9783527603978.mst0108.

@inbook{blank:2006,
    author = "Blank, Hubert",
    publisher = "John Wiley \\& Sons, Ltd",
    isbn = "9783527603978",
    title = "Nonoxide Ceramic Nuclear Fuels",
    booktitle = "Materials Science and Technology",
    chapter = "",
    pages = "",
    doi = "10.1002/9783527603978.mst0108",
    url = "https://onlinelibrary.wiley.com/doi/abs/10.1002/9783527603978.mst0108",
    year = "2006"
}

Hj Matzke. Science of advanced LMFBR fuels. Elsevier Science Pub. Co. Inc, 1986.

Timm Preusser. Modeling of carbide fuel rods. Nuclear Technology, 57(3):343–371, 1982. URL: https://doi.org/10.13182/NT82-A26303, arXiv:https://doi.org/10.13182/NT82-A26303, doi:10.13182/NT82-A26303.

@article{preusser:1982,
    author = "Preusser, Timm",
    title = "Modeling of Carbide Fuel Rods",
    journal = "Nuclear Technology",
    volume = "57",
    number = "3",
    pages = "343-371",
    year = "1982",
    publisher = "Taylor \& Francis",
    doi = "10.13182/NT82-A26303",
    URL = "https://doi.org/10.13182/NT82-A26303",
    eprint = "https://doi.org/10.13182/NT82-A26303"
}

H. Steiner. Das materialverhalten der karbidischen brennstoff. IMF-Bericht 229, PBSBericht 440 (1b), 1975.

@misc{steiner1975materialverhalten,
    author = "Steiner, H.",
    title = "Das materialverhalten der karbidischen brennstoff",
    year = "1975",
    howpublished = "IMF-Bericht 229, PBSBericht 440 (1b)"
}

(test/tests/mc_thermal/nonad.i)

# This test is to verify MCThermal appropriately creates the specific heat and thermal conductivity
# materials for use in HeatConduction and HeatConductionTimeDerivative

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
  []
[]

[Variables]
  [temp]
    initial_condition = 400
  []
[]

[Kernels]
  [dt]
    type = HeatConductionTimeDerivative
    variable = temp
    density_name = 1
  []
  [diff]
    type = HeatConduction
    variable = temp
  []
[]

[Materials]
  [thermal]
    type = MCThermal
    temperature = temp
    X_Pu = 0.1
    porosity = 0.1
    outputs = all
  []
[]

[BCs]
  [bottom]
    type = DirichletBC
    variable = temp
    value = 400
    boundary = bottom
  []
  [top]
    type = FunctionDirichletBC
    variable = temp
    boundary = top
    function = '400 + t * 100'
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  dt = 1
  num_steps = 10
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [thcond]
    type = ElementAverageValue
    variable = thermal_conductivity
  []
  [spheat]
    type = ElementAverageValue
    variable = specific_heat
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/mc_thermal/nonad.i)

# This test is to verify MCThermal appropriately creates the specific heat and thermal conductivity
# materials for use in HeatConduction and HeatConductionTimeDerivative

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
  []
[]

[Variables]
  [temp]
    initial_condition = 400
  []
[]

[Kernels]
  [dt]
    type = HeatConductionTimeDerivative
    variable = temp
    density_name = 1
  []
  [diff]
    type = HeatConduction
    variable = temp
  []
[]

[Materials]
  [thermal]
    type = MCThermal
    temperature = temp
    X_Pu = 0.1
    porosity = 0.1
    outputs = all
  []
[]

[BCs]
  [bottom]
    type = DirichletBC
    variable = temp
    value = 400
    boundary = bottom
  []
  [top]
    type = FunctionDirichletBC
    variable = temp
    boundary = top
    function = '400 + t * 100'
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  dt = 1
  num_steps = 10
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [thcond]
    type = ElementAverageValue
    variable = thermal_conductivity
  []
  [spheat]
    type = ElementAverageValue
    variable = specific_heat
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/mc_thermal/preusser_spheat_exact.i)

# Take ADMCThermal materials out for a spin

[Problem]
  solve = false
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[AuxVariables]
  [temp]
    initial_condition = 500.0
  []
[]

[AuxKernels]
  [temp_aux]
    type = FunctionAux
    variable = temp
    function = 'temperature_ramp'
  []
[]

[Functions]
  [temperature_ramp]
    type = PiecewiseLinear
    x = '0 5 10'
    y = '500 1000 500'
  []
  [X_Pu_ramp]
    type = ParsedFunction
    expression = '0.02 * t + 0.05'
  []
  [spheat_func]
    type = ParsedFunction
    symbol_names = 'temp X_Pu A_U A_Pu A_C'
    symbol_values = 'temp_avg X_Pu_avg 0.2380289 0.244 0.0120107'
    expression = 'uc_spheat := 54.4 + 9.62e-3 * temp;
             upuc_spheat := 50.2 + 9.62e-3 * temp;
             atomic_mass := (1 - X_Pu) * A_U + A_Pu * X_Pu + A_C;
             ((upuc_spheat - uc_spheat) / 0.2 * X_Pu + uc_spheat) / atomic_mass'
  []
[]

[Materials]
  [thermal]
    type = ADMCThermal
    temperature = temp
    porosity = 0
    specific_heat_model = PREUSSER
    X_Pu = X_Pu
    outputs = all
  []
  [X_Pu]
    type = ADGenericFunctionMaterial
    prop_names = 'X_Pu'
    prop_values = 'X_Pu_ramp'
    outputs = all
  []
[]

[Executioner]
  type = Transient
  end_time = 10
  dt = 0.5
[]

[Postprocessors]
  [temp_avg]
    type = ElementAverageValue
    variable = temp
  []
  [X_Pu_avg]
    type = ElementAverageValue
    variable = X_Pu
  []

  [spheat_avg]
    type = ElementAverageValue
    variable = specific_heat
  []
  [spheat_exact]
    type = FunctionValuePostprocessor
    function = spheat_func
  []
  [spheat_fractional_diff]
    type = ParsedPostprocessor
    pp_names = 'spheat_exact spheat_avg'
    expression = '(spheat_exact - spheat_avg) / spheat_exact'
    outputs = none
  []
  [spheat_max_diff]
    type = TimeExtremeValue
    postprocessor = spheat_fractional_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

(test/tests/mc_thermal/thcond_porosity_exact.i)

# Take ADMCThermal materials out for a spin

[Problem]
  solve = false
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[Functions]
  [porosity_ramp]
    type = ParsedFunction
    expression = 't / 30'
  []
  [thcond_func]
    type = ParsedFunction
    symbol_names = 'porosity_avg'
    symbol_values = 'porosity_avg'
    expression = '20 * (1 - porosity_avg) / (1 + porosity_avg)'
  []
[]

[Materials]
  [thermal]
    type = ADMCThermal
    temperature = 500
    porosity = porosity
    thermal_conductivity_model = STEINER
    X_Pu = 0
    outputs = all
  []
  [porosity]
    type = ADGenericFunctionMaterial
    prop_names = 'porosity'
    prop_values = 'porosity_ramp'
    outputs = all
  []
[]

[Executioner]
  type = Transient
  end_time = 10
[]

[Postprocessors]
  [porosity_avg]
    type = ElementAverageValue
    variable = porosity
  []
  [thcond_avg]
    type = ElementAverageValue
    variable = thermal_conductivity
  []
  [thcond_exact]
    type = FunctionValuePostprocessor
    function = thcond_func
  []
  [tcond_fractional_diff]
    type = ParsedPostprocessor
    pp_names = 'thcond_exact thcond_avg'
    expression = '(thcond_exact - thcond_avg) / thcond_exact'
    outputs = none
  []
  [tcond_max_diff]
    type = TimeExtremeValue
    postprocessor = tcond_fractional_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

(test/tests/mc_thermal/steiner_thcond_exact.i)

# Take ADMCThermal materials out for a spin

[Problem]
  solve = false
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[AuxVariables]
  [temp]
    initial_condition = 500.0
  []
[]

[AuxKernels]
  [temp_aux]
    type = FunctionAux
    variable = temp
    function = 'temperature_ramp'
  []
[]

[Functions]
  [temperature_ramp]
    type = PiecewiseLinear
    x = '0 5 10'
    y = '500 1000 500'
  []
  [X_Pu_ramp]
    type = ParsedFunction
    expression = '0.02 * t + 0.05'
  []
  [thcond_func]
    type = ParsedFunction
    symbol_names = 'temp X_Pu uc_low_t uc_high_t upuc_low_t upuc_high_t'
    symbol_values = 'temp_avg X_Pu_avg 773 798 773 798'
    expression = 'c := temp - 773;
             uc_low := 20;
             uc_high := 20 + 1.3e-3 * c;
             uc_x := if(temp < uc_low_t, 0, if(temp > uc_high_t, 1, (temp - uc_low_t) / (uc_high_t - uc_low_t)));
             uc_smooth := uc_x^3 * (uc_x * (uc_x * 6.0 - 15.0) + 10.0);
             uc_thcond := (1 - uc_smooth) * uc_low + uc_smooth * uc_high;
             upuc_low := 12.6;
             upuc_high := 12.6 + 4.1e-3 * c;
             upuc_x := if(temp < upuc_low_t, 0, if(temp > upuc_high_t, 1, (temp - upuc_low_t) / (upuc_high_t - upuc_low_t)));
             upuc_smooth := upuc_x^3 * (upuc_x * (upuc_x * 6.0 - 15.0) + 10.0);
             upuc_thcond := (1 - upuc_smooth) * upuc_low + upuc_smooth * upuc_high;
             (upuc_thcond - uc_thcond) / 0.2 * X_Pu + uc_thcond'
  []
[]

[Materials]
  [thermal]
    type = ADMCThermal
    temperature = temp
    porosity = 0
    thermal_conductivity_model = STEINER
    X_Pu = X_Pu
    outputs = all
  []
  [X_Pu]
    type = ADGenericFunctionMaterial
    prop_names = 'X_Pu'
    prop_values = 'X_Pu_ramp'
    outputs = all
  []
[]

[Executioner]
  type = Transient
  end_time = 10
  dt = 0.5
[]

[Postprocessors]
  [temp_avg]
    type = ElementAverageValue
    variable = temp
  []
  [X_Pu_avg]
    type = ElementAverageValue
    variable = X_Pu
  []

  [thcond_avg]
    type = ElementAverageValue
    variable = thermal_conductivity
  []
  [thcond_exact]
    type = FunctionValuePostprocessor
    function = thcond_func
  []
  [tcond_fractional_diff]
    type = ParsedPostprocessor
    pp_names = 'thcond_exact thcond_avg'
    expression = '(thcond_exact - thcond_avg) / thcond_exact'
    outputs = none
  []
  [tcond_max_diff]
    type = TimeExtremeValue
    postprocessor = tcond_fractional_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

(test/tests/mc_thermal/matzke_thcond_exact.i)

# Take ADMCThermal materials out for a spin

[Problem]
  solve = false
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[AuxVariables]
  [temp]
    initial_condition = 500.0
  []
[]

[AuxKernels]
  [temp_aux]
    type = FunctionAux
    variable = temp
    function = 'temperature_ramp'
  []
[]

[Functions]
  [temperature_ramp]
    type = PiecewiseLinear
    x = '0 5 10'
    y = '500 1000 500'
  []
  [X_Pu_ramp]
    type = ParsedFunction
    expression = '0.02 * t + 0.05'
  []
  [thcond_func]
    type = ParsedFunction
    symbol_names = 'temp X_Pu uc_low_t uc_high_t upuc_low_t upuc_high_t'
    symbol_values = 'temp_avg X_Pu_avg 948 998 748 798'
    expression = 'c := temp - 273.15;
             uc_low := 21.7 - 3.04e-3 * c + 3.61e-6 * c * c;
             uc_high := 20.2 + 1.48e-3 * c;
             uc_x := if(temp < uc_low_t, 0, if(temp > uc_high_t, 1, (temp - uc_low_t) / (uc_high_t - uc_low_t)));
             uc_smooth := uc_x^3 * (uc_x * (uc_x * 6.0 - 15.0) + 10.0);
             uc_thcond := (1 - uc_smooth) * uc_low + uc_smooth * uc_high;
             upuc_low := 17.5 - 5.65e-3 * c + 8.14e-6 * c * c;
             upuc_high := 12.76 + 8.71e-3 * c - 1.88e-6 * c * c;
             upuc_x := if(temp < upuc_low_t, 0, if(temp > upuc_high_t, 1, (temp - upuc_low_t) / (upuc_high_t - upuc_low_t)));
             upuc_smooth := upuc_x^3 * (upuc_x * (upuc_x * 6.0 - 15.0) + 10.0);
             upuc_thcond := (1 - upuc_smooth) * upuc_low + upuc_smooth * upuc_high;
             (upuc_thcond - uc_thcond) / 0.2 * X_Pu + uc_thcond'
  []
[]

[Materials]
  [thermal]
    type = ADMCThermal
    temperature = temp
    porosity = 0
    thermal_conductivity_model = MATZKE
    X_Pu = X_Pu
    outputs = all
  []
  [X_Pu]
    type = ADGenericFunctionMaterial
    prop_names = 'X_Pu'
    prop_values = 'X_Pu_ramp'
    outputs = all
  []
[]

[Executioner]
  type = Transient
  end_time = 10
  dt = 0.5
[]

[Postprocessors]
  [temp_avg]
    type = ElementAverageValue
    variable = temp
  []
  [X_Pu_avg]
    type = ElementAverageValue
    variable = X_Pu
  []

  [thcond_avg]
    type = ElementAverageValue
    variable = thermal_conductivity
  []
  [thcond_exact]
    type = FunctionValuePostprocessor
    function = thcond_func
  []
  [tcond_fractional_diff]
    type = ParsedPostprocessor
    pp_names = 'thcond_exact thcond_avg'
    expression = '(thcond_exact - thcond_avg) / thcond_exact'
    outputs = none
  []
  [tcond_max_diff]
    type = TimeExtremeValue
    postprocessor = tcond_fractional_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

(test/tests/mc_thermal/ad.i)

# This test is to verify MCThermal appropriately creates the specific heat and thermal conductivity
# materials for use in HeatConduction and HeatConductionTimeDerivative

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
  []
[]

[Variables]
  [temp]
    initial_condition = 400
  []
[]

[Kernels]
  [dt]
    type = ADHeatConductionTimeDerivative
    variable = temp
    density_name = 1
  []
  [diff]
    type = ADHeatConduction
    variable = temp
  []
[]

[Materials]
  [thermal]
    type = ADMCThermal
    temperature = temp
    X_Pu = 0.1
    porosity = 0.1
    outputs = all
  []
[]

[BCs]
  [bottom]
    type = DirichletBC
    variable = temp
    value = 400
    boundary = bottom
  []
  [top]
    type = FunctionDirichletBC
    variable = temp
    boundary = top
    function = '400 + t * 100'
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  dt = 1
  num_steps = 10
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [thcond]
    type = ElementAverageValue
    variable = thermal_conductivity
  []
  [spheat]
    type = ElementAverageValue
    variable = specific_heat
  []
[]

[Outputs]
  exodus = true
[]

(assessment/carbide/EBRII/WSA32/analysis/base.i)

initial_fuel_density = 13630.0

[GlobalParams]
  density = ${initial_fuel_density}
  order = SECOND
  family = LAGRANGE
  energy_per_fission = 3.2e-11 # J/fission
  volumetric_locking_correction = false
  displacements = 'disp_x disp_y'
  value_range_behavior = IGNORE
[]

[Problem]
  type = ReferenceResidualProblem
  reference_vector = 'ref'
  extra_tag_vectors = 'ref'
  group_variables = 'disp_x disp_y'
[]

[Mesh]
  coord_type = RZ
  use_displaced_mesh = false
  [smeared_pellet_mesh]
    type = FuelPinMeshGenerator
    clad_thickness = 0.508e-3
    pellet_outer_radius = 4.301e-3
    pellet_height = 343e-3
    clad_top_gap_height = 43.5e-3
    clad_gap_width = 0.145e-3
    bottom_clad_height = 8e-3
    top_clad_height = 8e-3
    clad_bot_gap_height = 0.2e-3 # unknown
    clad_mesh_density = customize
    pellet_mesh_density = customize
    nx_p = 6
    ny_p = 260
    nx_c = 4
    ny_c = 260
    ny_cu = 3
    ny_cl = 3
    pellet_quantity = 1
    elem_type = QUAD8
  []
  patch_size = 30
  patch_update_strategy = auto
  partitioner = centroid
  centroid_partitioner_direction = y
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [temp]
    initial_condition = 350
  []
[]

[AuxVariables]
  [contact_pressure]
  []
  [layered_side_contact_pressure]
    block = pellet
    order = CONSTANT
    family = MONOMIAL
  []
  [pellet_wall_temp]
    order = CONSTANT
    family = MONOMIAL
  []
  [radial_heat_flux]
    order = CONSTANT
    family = MONOMIAL
  []
  [clad_inner_wall_temp]
    order = CONSTANT
    family = MONOMIAL
  []
  [volumetric_strain]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '0 1e5   70e6 70100000' # unknown, final burnup was 12.2 %
    y = '0 1     1    0'
  []
  [axial_peaking_factors]
    type = PiecewiseLinear
    axis = y
    x = '0    46e-3 111e-3 200e-3 248e-3 313e-3 352e-3'
    y = '86e3 86e3  97e3   100e3  96e3   87e3   87e3'
  []
  [fission_func]
    type = ParsedFunction
    symbol_names = 'axial_peaking_factors power_history'
    symbol_values = 'axial_peaking_factors power_history'
    expression = 'linear_heating_rate := axial_peaking_factors * power_history;
             pellet_cross_sectional_area := 3.14159 * pow( 4.301e-3, 2 );
             volumetric_power := linear_heating_rate / pellet_cross_sectional_area;
             energy_per_fission := 3.2e-11;
             volumetric_power / energy_per_fission'
  []
  [burnup_func] # Burnup func is needed because BurnupAux doesn't currentl export an ADMaterialProperty.
    type = PiecewiseLinear
    x = '0 1e5   70e6  70100000'
    y = '0 0     0.122 0.122'
  []

  [coolant_press_ramp]
    type = PiecewiseLinear
    x = '0       70100000'
    y = '0.151e6 0.151e6' # unknown, Na coolant
  []
  [coolant_inlet_temp]
    type = PiecewiseLinear
    x = '0   1e5 70e6 70100000'
    y = '350 644 644  350'
  []
  [coolant_outlet_temp]
    type = PiecewiseLinear
    x = '0   1e5 70e6 70100000'
    y = '350 746 746  350'
  []
  [coolant_wall_temp]
    type = ParsedFunction
    symbol_values = 'coolant_inlet_temp coolant_outlet_temp'
    symbol_names = 'coolant_inlet_temp coolant_outlet_temp'
    expression = 'rod_length := 343e-3 + 43.5e-3 + 2 * 8e-3;
             delta_temp := coolant_outlet_temp - coolant_inlet_temp;
             interpolated_temp := coolant_inlet_temp + y * delta_temp / rod_length;
             max( min( interpolated_temp, coolant_outlet_temp ), coolant_inlet_temp )'
  []
  [radial_heat_flux]
    type = ParsedFunction
    symbol_values = 'coolant_wall_temp power_history axial_peaking_factors'
    symbol_names = 'coolant_wall_temp power_history axial_peaking_factors'
    expression = 'linear_heating_rate := axial_peaking_factors * power_history;
             pellet_radius := 4.301e-3;
             pellet_circumference := 3.14159 * pellet_radius * 2;
             linear_heating_rate / pellet_circumference'
  []
  [clad_inner_wall_temp]
    type = ParsedFunction
    symbol_values = 'radial_heat_flux coolant_wall_temp'
    symbol_names = 'radial_heat_flux coolant_wall_temp'
    expression = 'conductivity_316ss := 22;
             clad_thickness := 0.51e-3;
             coolant_wall_temp + radial_heat_flux / conductivity_316ss * clad_thickness'
  []
  [pellet_wall_temp]
    type = ParsedFunction
    symbol_values = 'radial_heat_flux coolant_wall_temp clad_inner_wall_temp'
    symbol_names = 'radial_heat_flux coolant_wall_temp clad_inner_wall_temp'
    expression = 'conductivity_gap := 1;
             gap_thickness := 0.145e-3;
             clad_inner_wall_temp + radial_heat_flux / conductivity_gap * gap_thickness'
  []

  [contact_pressure_func]
    type = ConstantFunction
    value = 12e6
  []
  [plenum_pressure]
    type = ConstantFunction
    value = 12.4e6 # initial fill
  []

  [porosity_ramp]
    type = PiecewiseLinear
    x = '0    1e5  70e6  70100000'
    y = '0.14 0.14 0.145 0.145' # Pellets start at 88.1% TD, but no data on final porosity.
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [fuel]
    block = pellet
    strain = SMALL
    incremental = true
    extra_vector_tags = 'ref'
    eigenstrain_names = 'fuel_thermal_expansion fuel_swelling'
    use_automatic_differentiation = true
  []
  [clad]
    block = clad
    strain = SMALL
    incremental = true
    extra_vector_tags = 'ref'
    use_automatic_differentiation = true
  []
[]

[Kernels]
  [gravity]
    type = ADGravity
    variable = disp_y
    value = -9.81
    extra_vector_tags = 'ref'
  []
  [heat]
    type = ADHeatConduction
    variable = temp
    extra_vector_tags = 'ref'
  []
  [heat_ie]
    type = ADHeatConductionTimeDerivative
    variable = temp
    extra_vector_tags = 'ref'
  []
  [heat_source]
    type = ADFissionRateHeatSource
    variable = temp
    block = pellet
    fission_rate = fission_rate
    extra_vector_tags = 'ref'
  []
[]

[AuxKernels]
  [contact_pressure_aux]
    type = FunctionAux
    variable = contact_pressure
    function = contact_pressure_func
  []
  [layered_side_contact_pressure]
    type = SpatialUserObjectAux
    variable = layered_side_contact_pressure
    user_object = layered_side_contact_pressure
    block = pellet
  []

  [pellet_wall_temp]
    type = FunctionAux
    variable = pellet_wall_temp
    function = pellet_wall_temp
  []
  [radial_heat_flux]
    type = FunctionAux
    variable = radial_heat_flux
    function = radial_heat_flux
  []
  [clad_inner_wall_temp]
    type = FunctionAux
    variable = clad_inner_wall_temp
    function = clad_inner_wall_temp
  []
  [volumetric_strain]
    type = ADRankTwoScalarAux
    rank_two_tensor = total_strain
    variable = volumetric_strain
    scalar_type = VolumetricStrain
  []
[]

[BCs]
  [no_x_all]
    type = DirichletBC
    variable = disp_x
    boundary = 12
    value = 0.0
  []
  [no_y_fuel]
    type = DirichletBC
    variable = disp_y
    boundary = 20
    value = 0.0
  []
  [no_y_clad]
    type = DirichletBC
    variable = disp_y
    boundary = 1
    value = 0.0
  []
  [Pressure]
    [coolantPressure]
      boundary = '1 2 3'
      function = coolant_press_ramp
    []
    [plenumPressure]
      boundary = 9
      function = plenum_pressure
    []
  []
  [clad_outer_temp]
    type = FunctionDirichletBC
    boundary = '1 2 3'
    function = coolant_wall_temp
    variable = temp
  []
  [pellet_to_clad_cooling]
    type = FunctionDirichletBC
    boundary = 10
    function = pellet_wall_temp
    variable = temp
  []
[]

[Materials]
  [fission]
    type = ADGenericFunctionMaterial
    prop_names = fission_rate
    prop_values = fission_func
    outputs = all
  []
  [burnup]
    type = ADGenericFunctionMaterial
    block = pellet
    prop_names = burnup
    prop_values = burnup_func
    outputs = all
  []
  [fuel_thermal]
    block = pellet
    type = ADMCThermal
    temperature = temp
    porosity = porosity
    X_Pu = 0.0
    outputs = all
  []
  [fuel_porosity]
    block = pellet
    type = ADGenericFunctionMaterial
    prop_names = porosity
    prop_values = porosity_ramp
    outputs = all
  []
  [fuel_elasticity_tensor]
    block = pellet
    type = ADMCElasticityTensor
    temperature = temp
    porosity = porosity
    output_properties = 'youngs_modulus poissons_ratio'
    outputs = all
  []
  [fuel_thermal_expansion]
    block = pellet
    type = ADMCThermalExpansionEigenstrain
    eigenstrain_name = fuel_thermal_expansion
    stress_free_temperature = 350
    temperature = temp
  []
  [fuel_creep]
    block = pellet
    type = ADMCCreepUpdate
    temperature = temp
    fission_rate = fission_rate
    outputs = all
  []
  [fuel_swelling]
    block = pellet
    type = ADMCVolumetricSwellingEigenstrain
    eigenstrain_name = fuel_swelling
    temperature = temp
    porosity = porosity
    contact_pressure = layered_side_contact_pressure
    burnup = burnup
    outputs = all
  []
  [fuel_radial_return_stress]
    block = pellet
    type = ADComputeMultipleInelasticStress
    inelastic_models = 'fuel_creep'
  []
  [fuel_density]
    block = pellet
    type = ADStrainAdjustedDensity
    strain_free_density = ${initial_fuel_density}
  []

  [clad_elasticity_tensor]
    block = clad
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 190e9
    poissons_ratio = 0.265
  []
  [clad_stress]
    block = clad
    type = ADComputeLinearElasticStress
  []
  [clad_thermal]
    block = clad
    type = ADSS316Thermal
    temperature = temp
  []
  [clad_density]
    block = clad
    type = ADStrainAdjustedDensity
    strain_free_density = 8000
  []
[]

[UserObjects]
  [layered_side_contact_pressure]
    # This reads the contact pressure at the pellet OD and propagates it inward so that it can
    # be used in ADMCVolumetricSwellingEigenstrain for inner elements as well.
    type = LayeredSideAverage
    direction = y
    num_layers = 100
    variable = contact_pressure
    execute_on = linear
    boundary = 10
  []
[]

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

[Dampers]
  [limitT]
    type = MaxIncrement
    max_increment = 100.0
    variable = temp
  []
  [limitX]
    type = MaxIncrement
    max_increment = 1e-5
    variable = disp_x
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 50
  l_tol = 8e-3
  nl_max_its = 15
  nl_rel_tol = 1e-4
  nl_abs_tol = 1e-10

  start_time = -200
  n_startup_steps = 1
  end_time = 70100000
  dtmin = 1
  dtmax = 2e6

  [Quadrature]
    order = fifth
    side_order = seventh
  []

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 2e2
    time_t = ' 0   1e5 70e6 70100000'
    time_dt = '1e2 1e2 1e2  1e2'
    optimal_iterations = 10
    growth_factor = 2
    cutback_factor = 0.5
    iteration_window = 1
    linear_iteration_ratio = 100
  []
[]

[Postprocessors]
  [_dt]
    type = TimestepSize
  []
  [num_lin_it]
    type = NumLinearIterations
  []
  [num_nonlin_it]
    type = NumNonlinearIterations
  []
  [burnup]
    block = pellet
    type = ADElementAverageMaterialProperty
    mat_prop = burnup
  []
  [FCT]
    type = NodalExtremeValue
    boundary = 12
    variable = temp
    execute_on = 'initial timestep_end'
  []
  [volumetric_strain]
    type = ElementAverageValue
    block = pellet
    variable = volumetric_strain
  []
  [pellet_outer_disp_x]
    type = NodalExtremeValue
    boundary = 10
    variable = disp_x
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  color = true
  exodus = true
  perf_graph = true
  csv = true
  [console]
    type = Console
    max_rows = 25
    time_step_interval =  1
    output_linear = true
  []
  [chkfile]
    type = CSV
    hide = 'num_nonlin_it num_lin_it _dt'
  []
[]

Description
Specific Heat Capacity
Example Input Syntax
Input Parameters
Input Files
References