ZrThermal

Computes the thermal conductivity and the specific heat, under constant pressure, for pure zirconium used as a liner in BWRs and plate fuel.

Description

The ZrThermal model computes the thermal conductivity and specific heat capacity of pure zirconium.

Thermal Conductivity

The thermal conductivity $var element = document.getElementById("moose-equation-d7812337-45ad-4e3b-8f1f-1e64e941a928");katex.render("k", element, {displayMode:false,throwOnError:false});$ (W/m-K) of pure zirconium is given by Fink and Leibowitz (1995) and Rabin et al. (2020). Three options are available.

The Fink option computes $var element = document.getElementById("moose-equation-0453ba0d-7bf5-416a-abe0-722a418083e8");katex.render("k", element, {displayMode:false,throwOnError:false});$ as: $var element = document.getElementById("moose-equation-d11122b2-d9cb-4e9e-b63f-91cc5daf46b2");katex.render(" k = 8.8527 + 7.0820 \\times 10^{-3} T + 2.5329 \\times 10^{-6} T^2 + \\frac{2.9918 \\times 10^{3}}{T}", element, {displayMode:true,throwOnError:false});$
where 298 K $var element = document.getElementById("moose-equation-d7b83675-eb95-42b8-b30d-1668c35febaf");katex.render("\\leq T \\leq", element, {displayMode:false,throwOnError:false});$ 2000 K is the temperature.
The USHPRR option uses a piecewise linear relationship between temperature and thermal conductivity. For details, see p. 451 in Rabin et al. (2020).
The USHPRRFit option uses a smoothed fit of the piecewise linear relationship in the USHPRR option. This preserves the character of the USHPRR data while providing a smooth fit that provides better derivative information.

Specific Heat Capacity

The specific heat $var element = document.getElementById("moose-equation-283c169e-3fc8-46af-8f7c-7a4e294fab06");katex.render("c_P", element, {displayMode:false,throwOnError:false});$ (J/kg/K) of pure zirconium is given by Fink and Leibowitz (1995) and Rabin et al. (2020). Three options are available.

For the Fink option, the specific heat correlation depends upon whether the zirconium is in the $var element = document.getElementById("moose-equation-9b359f5e-74e5-411a-b3bd-ac24fb165f99");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ (298 $var element = document.getElementById("moose-equation-3eb0b9e4-c2a9-4af1-9c93-29b42ddf3148");katex.render("\\leq T \\leq", element, {displayMode:false,throwOnError:false});$ 1139 K) or $var element = document.getElementById("moose-equation-70bc976e-90c2-4923-8297-c4fb1d70bab3");katex.render("\\beta", element, {displayMode:false,throwOnError:false});$ (1139 $var element = document.getElementById("moose-equation-2568ca56-dd85-4c96-8142-372da4f08f3c");katex.render("<", element, {displayMode:false,throwOnError:false});$ T $var element = document.getElementById("moose-equation-318029d6-26d7-4c33-acf9-80fa84d0f456");katex.render("\\leq", element, {displayMode:false,throwOnError:false});$ 2128 K) phase.
$var element = document.getElementById("moose-equation-43ffb035-5536-42b5-91c6-8431de251335");katex.render(" c_P = \\begin{cases} 24.1618 + 8.75582 \\times 10^{-3}T - \\frac{69942}{T^2} & 298 \\text{K} < T < 1139 \\text{K} \\\\ 25.607406 + 6.80168 \\times 10^{-4} T + 5.837384 \\times 10^{-8} T^2 + 9.13714728 \\times 10^{-10} T^3 - \\frac{50466}{T^2} & 1139 \\text{K} \\leq T \\leq 2128 \\text{K} \\end{cases}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-22cb7984-7183-42aa-81bb-290daf89a477");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature in K.
The USHPRR option uses a piecewise linear relationship between temperature and specific heat. For details, see p. 450 in Rabin et al. (2020).
The USHPRRFit option uses a smoothed fit of the piecewise linear relationship in the USHPRR option. This preserves the character of the USHPRR data while providing a smooth fit that provides better derivative information.

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [thermalZr]
    type = ZrThermal<<<{"description": "Computes the thermal conductivity and the specific heat, under constant pressure, for pure zirconium used as a liner in BWRs and plate fuel.", "href": "ZrThermal.html"}>>>
    temperature<<<{"description": "Coupled temperature"}>>> = temperature
  []
  [thermalZr_USHPRR]
    type = ZrThermal<<<{"description": "Computes the thermal conductivity and the specific heat, under constant pressure, for pure zirconium used as a liner in BWRs and plate fuel.", "href": "ZrThermal.html"}>>>
    temperature<<<{"description": "Coupled temperature"}>>> = temperature
    thermal_conductivity_model<<<{"description": "The model to use for specific heat: Fink USHPRR USHPRRFit"}>>> = USHPRR
    specific_heat_model<<<{"description": "The model to use for specific heat: Fink USHPRR USHPRRFit"}>>> = USHPRR
    base_name<<<{"description": "Optional parameter that allows the user to define multiple material systems on the same block, i.e. for multiple phases"}>>> = USHPRR
  []
  [thermalZr_USHPRRFit]
    type = ZrThermal<<<{"description": "Computes the thermal conductivity and the specific heat, under constant pressure, for pure zirconium used as a liner in BWRs and plate fuel.", "href": "ZrThermal.html"}>>>
    temperature<<<{"description": "Coupled temperature"}>>> = temperature
    thermal_conductivity_model<<<{"description": "The model to use for specific heat: Fink USHPRR USHPRRFit"}>>> = USHPRRFit
    specific_heat_model<<<{"description": "The model to use for specific heat: Fink USHPRR USHPRRFit"}>>> = USHPRRFit
    base_name<<<{"description": "Optional parameter that allows the user to define multiple material systems on the same block, i.e. for multiple phases"}>>> = USHPRRFit
  []
  [density]
    type = ParsedMaterial<<<{"description": "Parsed expression Material.", "href": "ParsedMaterial.html"}>>>
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 0
    property_name<<<{"description": "Name of the parsed material property"}>>> = density
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = 5800.0
  []
[]

Input Parameters

temperatureCoupled temperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled temperature

Required Parameters

base_nameOptional parameter that allows the user to define multiple material systems on the same block, i.e. for multiple phases
C++ Type:std::string
Controllable:No
Description:Optional parameter that allows the user to define multiple material systems on the same block, i.e. for multiple phases
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.
silence_warningsFalseWhether to silence correlation out of bound warnings
Default:False
C++ Type:bool
Controllable:No
Description:Whether to silence correlation out of bound warnings
specific_heat_modelFinkThe model to use for specific heat: Fink USHPRR USHPRRFit
Default:Fink
C++ Type:MooseEnum
Options:Fink, USHPRR, USHPRRFit
Controllable:No
Description:The model to use for specific heat: Fink USHPRR USHPRRFit
thermal_conductivity_modelFinkThe model to use for specific heat: Fink USHPRR USHPRRFit
Default:Fink
C++ Type:MooseEnum
Options:Fink, USHPRR, USHPRRFit
Controllable:No
Description:The model to use for specific heat: Fink USHPRR USHPRRFit

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

specific_heat_scale_factor1Optional scaling factor applied to the overall specific heat.
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Optional scaling factor applied to the overall specific heat.
thermal_conductivity_scale_factor1Optional scaling factor applied to the overall thermal conductivity.
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Optional scaling factor applied to the overall thermal conductivity.

Advanced: Scaling Factors Parameters

References

J. K. Fink and L. Leibowitz. Thermal conductivity of zirconium. Journal of Nuclear Materials, 226(1–2):44–50, 1995. doi:10.1016/0022-3115(95)00110-7.

@article{fink1995,
    author = "Fink, J. K. and Leibowitz, L.",
    title = "Thermal conductivity of zirconium",
    journal = "Journal of Nuclear Materials",
    volume = "226",
    number = "1--2",
    pages = "44--50",
    doi = "10.1016/0022-3115(95)00110-7",
    year = "1995"
}

B. Rabin, M. Meyer, J. Cole, I. Glagolenko, W. Jones, J-F. Jue, D. Keiser Jr, C. Miller, G. Moore, H. Ozaltun, F. Rice, A. Robinson, J. Smith, D. Wachs, W. Williams, N. Woolstenhulme, G. Hofman, and Y. Kim. Preliminary report on U-Mo monolithic fuel for research reactors. Technical Report INL EXT-17-40975, Idaho National Laboratory, 4 2020.

@techreport{EXT-17-40975R4,
    author = "Rabin, B. and Meyer, M. and Cole, J. and Glagolenko, I. and Jones, W. and Jue, J-F. and Jr, D. Keiser and Miller, C. and Moore, G. and Ozaltun, H. and Rice, F. and Robinson, A. and Smith, J. and Wachs, D. and Williams, W. and Woolstenhulme, N. and Hofman, G. and Kim, Y.",
    title = "Preliminary Report on {U-Mo} Monolithic Fuel for Research Reactors",
    institution = "Idaho National Laboratory",
    number = "INL EXT-17-40975",
    month = "4",
    year = "2020"
}

(test/tests/thermalZr/thermal.i)

# The temperature is ramped on all faces of the cube from 300 K to 2100 K
# over 18000 s.
#
# Thermal conductivity and specific heat capacity are computed using the models
# by Fink and Leibowitz (1995) for pure zirconium.
#
# The thermal conductivity and specific heat computed by BISON for the
# temperatures shown below and compared with analytical solutions.
# The results are as follows:
#
#                 Zr k           Zr k             Zr Cp           Zr Cp
# Temp(K)     BISON(W/m-K)  analytical(W/m-K)  BISON(J/kg-K)  analytical(J/kg-K)
#  300          21.1779         21.1779          285.138         285.138
#  600          19.0001         19.0001          320.322         320.322
#  900          20.6024         20.6024          350.299         350.299
# 1200          23.4916         23.4916          307.502         307.502
# 1500          27.1693         27.1693          326.892         326.892
# 1800          31.4690         31.4690          354.447         354.447
# 2100          36.3197         36.3197          391.823         391.823

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    dim = 2
  []
[]

[Variables]
  [temperature]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
  [temperatureUSHPRR]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
  [temperatureUSHPRRFit]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temperature
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temperature
  []
  [heatUSHPRR]
    type = HeatConduction
    variable = temperatureUSHPRR
  []
  [heat_ieUSHPRR]
    type = HeatConductionTimeDerivative
    variable = temperatureUSHPRR
  []
  [heatUSHPRRFit]
    type = HeatConduction
    variable = temperatureUSHPRRFit
  []
  [heat_ieUSHPRRFit]
    type = HeatConductionTimeDerivative
    variable = temperatureUSHPRRFit
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 18000'
    y = '300 2100'
  []
[]

[BCs]
  [VariableT]
    type = FunctionDirichletBC
    boundary = 'left right bottom top'
    variable = temperature
    function = temp_ramp
  []
  [VariableTUSHPRR]
    type = FunctionDirichletBC
    boundary = 'left right bottom top'
    variable = temperatureUSHPRR
    function = temp_ramp
  []
  [VariableTUSHPRRFit]
    type = FunctionDirichletBC
    boundary = 'left right bottom top'
    variable = temperatureUSHPRRFit
    function = temp_ramp
  []
[]

[Materials]
  [thermalZr]
    type = ZrThermal
    temperature = temperature
  []
  [thermalZr_USHPRR]
    type = ZrThermal
    temperature = temperature
    thermal_conductivity_model = USHPRR
    specific_heat_model = USHPRR
    base_name = USHPRR
  []
  [thermalZr_USHPRRFit]
    type = ZrThermal
    temperature = temperature
    thermal_conductivity_model = USHPRRFit
    specific_heat_model = USHPRRFit
    base_name = USHPRRFit
  []
  [density]
    type = ParsedMaterial
    block = 0
    property_name = density
    expression = 5800.0
  []
[]

[Executioner]
   type = Transient
   solve_type = PJFNK

   petsc_options = '-snes_ksp_ew'
   petsc_options_iname = '-ksp_gmres_restart'
   petsc_options_value = '101'

   line_search = 'none'

   l_max_its = 100
   nl_max_its = 100
   nl_rel_tol = 1e-8
   nl_abs_tol = 1e-10
   l_tol = 1e-5
   start_time = 0.0
   num_steps = 18
   dt = 1000
[]

[Postprocessors]
  [avg_temperature]
    type = SideAverageValue
    boundary = 'left right top bottom'
    execute_on = 'initial timestep_end'
    variable = temperature
  []
  [avg_th_cond]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [avg_th_cond_USHPRR]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRR_thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [avg_th_cond_USHPRRFit]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRRFit_thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [avg_specific_heat]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = specific_heat
    execute_on = 'initial timestep_end'
  []
  [avg_specific_heat_USHPRR]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRR_specific_heat
    execute_on = 'initial timestep_end'
  []
  [avg_specific_heat_USHPRRFit]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRRFit_specific_heat
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  csv = true
[]

(examples/plate_fuel/plate_fuel_hypre.i)

#
# Plate fuel example problem.
#

[GlobalParams]
  order = FIRST
  family = LAGRANGE
  displacements = 'disp_x disp_y disp_z'
  volumetric_locking_correction = false
[]

[Mesh]
  coord_type = XYZ
  [plate]
    type = PlateMeshGenerator
    fuel_dimensions = '82.550e-3 19.050e-3 0.216e-3'
    liner_thickness = 0.025e-3
    cladding_thicknesses = '9.4615e-3 9.4615e-3 3.175e-3 3.175e-3 0.489e-3 0.489e-3'
    number_fuel_elements = '25 15 3'
    number_cladding_elements = '4 4 3 3 4 4'
    ### For a more refined mesh (40 processors):
    # number_fuel_elements = '108 30 5'
    # number_cladding_elements = '15 15 5 5 8 8'
    ###
    number_liner_elements = 1
  []
  [translate_to_origin]
    type = TransformGenerator
    input = plate
    transform = translate_min_origin
  []
  [add_nodesets]
    type = ExtraNodesetGenerator
    input = translate_to_origin
    coord = '0 0 0; 0.101473 0 0; 0.101473 0.0254 0'
    new_boundary = 'left_bottom_back right_bottom_back right_top_back'
  []
[]

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

[Functions]
  [fission_density_fcn] # fis/m3
    type = PiecewiseLinear
    data_file = 'average_fission_density.csv'
    scale_factor = 1e27
    format = columns
  []
  [fluence_fcn] # n/m2
    type = PiecewiseLinear
    data_file = 'FLUENCE.csv'
    scale_factor = 1
    format = columns
  []
  [power_density]
    type = PiecewiseLinear
    data_file = 'POWER_DENSITY.csv' # W/cm3 to W/m3
    scale_factor = 1e6
    format = columns
  []
  [L2AR]
    type = ParsedFunction
    # fissprof_l2ar is the distribution from the L2AR tab: L2AR_ABAQUS at cell R64
    symbol_names = 'x_offset y_offset'
    symbol_values = '9.4615e-3 3.175e-3'
    expression = 'x0 := x-x_offset;
             y0 := y-y_offset;
             (4.669e-10*x0^5 - 8.657e-8*x0^4 + 5.877e-6*x0^3 - 0.0001962*x0^2 + 0.004373*x0 + 0.9446) * (5.047e-7*y0^6 - 2.974e-05*y0^5 + 0.0006881*y0^4 - 0.007883*y0^3 + 0.04726*y0^2 - 0.1458*y0 + 1.152)'
  []
  [power_history]
    type = CompositeFunction
    functions = 'power_density L2AR'
  []
[]

[Variables]
  [temperature]
    initial_condition = 294
  []
  [disp_x]
  []
  [disp_y]
  []
  [disp_z]
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temperature
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temperature
  []
  [heat_source]
    type = HeatSource
    block = fuel
    function = power_history
    variable = temperature
    extra_vector_tags = 'ref'
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  strain = FINITE
  temperature = temperature
  generate_output = 'stress_xx stress_yy stress_zz vonmises_stress
                     hydrostatic_stress elastic_strain_xx elastic_strain_yy
                     elastic_strain_zz strain_xx strain_yy strain_zz'
  [fuel_mechanics]
    block = fuel
    eigenstrain_names = 'fuel_thermal_strain'
    extra_vector_tags = 'ref'
  []
  [clad_mechanics]
    block = cladding
    eigenstrain_names = 'clad_thermal_strain'
    extra_vector_tags = ref
  []
  [liner_mechanics]
    block = liner
    eigenstrain_names = 'liner_thermal_strain'
    extra_vector_tags = ref
  []
[]

[BCs]
  [conv_BC_front]
    type = ConvectiveHeatFluxBC
    variable = temperature
    boundary = front
    T_infinity = 'tfilm_outer' # film temperature
    heat_transfer_coefficient = 'htc_outer'
    heat_transfer_coefficient_dT = 'dhtc_dT_outer'
  []
  [conv_BC_back]
    type = ConvectiveHeatFluxBC
    variable = temperature
    boundary = back
    T_infinity = 'tfilm_inner' # film temperature
    heat_transfer_coefficient = 'htc_inner'
    heat_transfer_coefficient_dT = 'dhtc_dT_inner'
  []
  [disp_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'left_bottom_back'
    value = 0.0
  []
  [disp_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'left_bottom_back right_bottom_back'
    value = 0.0
  []
  [disp_z]
    type = DirichletBC
    variable = disp_z
    boundary = 'left_bottom_back right_bottom_back right_top_back'
    value = 0.0
  []
[]

[Materials]
  [fission_density]
    type = GenericFunctionMaterial
    prop_names = fission_density
    prop_values = fission_density_fcn
  []
  [fluence]
    type = GenericFunctionMaterial
    prop_names = fast_neutron_fluence
    prop_values = fluence_fcn
    outputs = all
  []
  [tfilm_inner]
    type = ParsedMaterial
    property_name = tfilm_inner
    coupled_variables = temperature
    expression = '325.15'
  []
  [htc_inner]
    type = ParsedMaterial
    property_name = htc_inner
    coupled_variables = 'temperature'
    expression = '-3.3647*temperature^2 + 2744.5*temperature - 454016.0'
  []
  [dhtc_dT_inner]
    type = ParsedMaterial
    property_name = dhtc_dT_inner
    coupled_variables = 'temperature'
    expression = '-6.7294*temperature + 2744.5'
  []
  [tfilm_outer]
    type = ParsedMaterial
    property_name = tfilm_outer
    coupled_variables = temperature
    expression = '325.15'
  []
  [htc_outer]
    type = ParsedMaterial
    property_name = htc_outer
    coupled_variables = 'temperature'
    expression = '-2.8992*temperature^2 + 2354.8*temperature - 387945.0'
  []
  [dhtc_dT_outer]
    type = ParsedMaterial
    property_name = dhtc_dT_outer
    coupled_variables = 'temperature'
    expression = '-5.7984*temperature + 2354.8'
  []
  # fuel properties
  [fuel_thermal]
    type = U10MoThermal
    block = fuel
    temperature = temperature
    thermal_conductivity_model = USHPRR
    specific_heat_model = USHPRR
  []
  [fuel_elasticity]
    type = U10MoElasticityTensor
    block = fuel
    temperature = temperature
  []
  [fuel_thermal_expansion]
    type = U10MoThermalExpansionEigenstrain
    block = fuel
    temperature = temperature
    eigenstrain_name = fuel_thermal_strain
    stress_free_temperature = 298
    model_option = Rest
  []
  [fuel_stress]
    type = ComputeFiniteStrainElasticStress
    block = fuel
  []
  #  [fuel_swelling]
  #    type = U10MoVolumetricSwellingEigenstrain
  #    eigenstrain_name = fuel_swelling_strain
  #    fission_rate = 0
  #  []
  [fuel_density]
    type = ParsedMaterial
    block = fuel
    property_name = density
    coupled_variables = 'temperature'
    expression = '-0.884*temperature + 17391.0'
    outputs = exodus
  []
  # liner properties
  [liner_thermal]
    type = ZrThermal
    block = liner
    temperature = temperature
    thermal_conductivity_model = USHPRR
    specific_heat_model = USHPRR
  []
  [liner_elasticity]
    type = ZrElasticityTensor
    block = liner
    temperature = temperature
  []
  [liner_stress]
    type = ComputeFiniteStrainElasticStress
    block = liner
  []
  [liner_thermal_expansion]
    type = ZrThermalExpansionEigenstrain
    block = liner
    temperature = temperature
    eigenstrain_name = liner_thermal_strain
    stress_free_temperature = 293.0
    model_option = ASM
  []
  [liner_density]
    type = StrainAdjustedDensity
    block = liner
    strain_free_density = 6499.0 # 6.499e-9 ton/mm3
    outputs = exodus
  []
  # cladding properties
  [clad_thermal]
    type = Al6061Thermal
    temperature = temperature
    block = cladding
    thermal_conductivity_model = T6
    specific_heat_model = ASM
  []
  [clad_elasticity]
    type = Al6061ElasticityTensor
    block = cladding
    temperature = temperature
    fast_neutron_fluence = 0
    youngs_modulus_model = Kaufman
  []
  [clad_thermal_expansion]
    type = Al6061ThermalExpansionEigenstrain
    stress_free_temperature = 293.15
    temperature = temperature
    eigenstrain_name = clad_thermal_strain
    block = cladding
  []
  [clad_stress]
    type = ComputeFiniteStrainElasticStress
    block = cladding
  []
  [clad_density]
    type = ParsedMaterial
    block = cladding
    property_name = density
    coupled_variables = 'temperature'
    expression = '-0.0003*temperature^2 + 0.0966*temperature + 2690.4'
    outputs = exodus
  []
[]

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

[Executioner]
  type = Transient
  solve_type = 'Newton'

  petsc_options_iname = '-ksp_gmres_restart -pc_type -pc_hypre_type -pc_hypre_boomeramg_max_iter'
  petsc_options_value = '201                hypre    boomeramg      4'
  #petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  #petsc_options_value = 'lu superlu_dist'

  line_search = 'none'

  nl_rel_tol = 1e-6
  nl_abs_tol = 1e-8
  nl_max_its = 50

  l_tol = 1e-4
  l_max_its = 50

  start_time = 0.0
  end_time = 4586400
  num_steps = 2500

  dtmax = 86400

  [Predictor]
    type = SimplePredictor
    scale = 1
  []

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1800
    optimal_iterations = 12
    iteration_window = 2
    growth_factor = 2
    cutback_factor = .5
    timestep_limiting_function = power_density
    force_step_every_function_point = true
  []
[]

[Postprocessors]
  [average_fuel_T]
    type = ElementAverageValue
    block = fuel
    variable = temperature
    execute_on = 'initial timestep_end'
  []
  [fuelVolume]
    type = InternalVolume
    boundary = fuel_all
  []
  [totalVolume]
    type = InternalVolume
    boundary = all
  []
  [power_history]
    type = FunctionValuePostprocessor
    function = power_history
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  perf_graph = true
  csv = true
  exodus = true
[]

(test/tests/thermalZr/ad_thermal.i)

# The temperature is ramped on all faces of the cube from 300 K to 2100 K
# over 18000 s.
#
# Thermal conductivity and specific heat capacity are computed using the models
# by Fink and Leibowitz (1995) for pure zirconium.
#
# The thermal conductivity and specific heat computed by BISON for the
# temperatures shown below and compared with analytical solutions.
# The results are as follows:
#
#                 Zr k           Zr k             Zr Cp           Zr Cp
# Temp(K)     BISON(W/m-K)  analytical(W/m-K)  BISON(J/kg-K)  analytical(J/kg-K)
#  300          21.1779         21.1779          285.138         285.138
#  600          19.0001         19.0001          320.322         320.322
#  900          20.6024         20.6024          350.299         350.299
# 1200          23.4916         23.4916          307.502         307.502
# 1500          27.1693         27.1693          326.892         326.892
# 1800          31.4690         31.4690          354.447         354.447
# 2100          36.3197         36.3197          391.823         391.823

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    dim = 2
  []
[]

[Variables]
  [temperature]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
  [temperatureUSHPRR]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
  [temperatureUSHPRRFit]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
[]

[Kernels]
  [heat]
    type = ADHeatConduction
    variable = temperature
  []
  [heat_ie]
    type = ADHeatConductionTimeDerivative
    variable = temperature
  []
  [heatUSHPRR]
    type = ADHeatConduction
    variable = temperatureUSHPRR
  []
  [heat_ieUSHPRR]
    type = ADHeatConductionTimeDerivative
    variable = temperatureUSHPRR
  []
  [heatUSHPRRFit]
    type = ADHeatConduction
    variable = temperatureUSHPRRFit
  []
  [heat_ieUSHPRRFit]
    type = ADHeatConductionTimeDerivative
    variable = temperatureUSHPRRFit
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 18000'
    y = '300 2100'
  []
[]

[BCs]
  [VariableT]
    type = FunctionDirichletBC
    boundary = 'left right bottom top'
    variable = temperature
    function = temp_ramp
  []
  [VariableTUSHPRR]
    type = FunctionDirichletBC
    boundary = 'left right bottom top'
    variable = temperatureUSHPRR
    function = temp_ramp
  []
  [VariableTUSHPRRFit]
    type = FunctionDirichletBC
    boundary = 'left right bottom top'
    variable = temperatureUSHPRRFit
    function = temp_ramp
  []
[]

[Materials]
  [thermalZr]
    type = ADZrThermal
    temperature = temperature
  []
  [thermalZr_USHPRR]
    type = ADZrThermal
    temperature = temperature
    thermal_conductivity_model = USHPRR
    specific_heat_model = USHPRR
    base_name = USHPRR
  []
  [thermalZr_USHPRRFit]
    type = ADZrThermal
    temperature = temperature
    thermal_conductivity_model = USHPRRFit
    specific_heat_model = USHPRRFit
    base_name = USHPRRFit
  []
  [density]
    type = ADParsedMaterial
    block = 0
    property_name = density
    expression = 5800.0
  []
[]

[Executioner]
   type = Transient
   solve_type = NEWTON

   petsc_options = '-snes_ksp_ew'
   petsc_options_iname = '-ksp_gmres_restart'
   petsc_options_value = '101'

   line_search = 'none'

   l_max_its = 100
   nl_max_its = 100
   nl_rel_tol = 1e-8
   nl_abs_tol = 1e-10
   l_tol = 1e-5
   start_time = 0.0
   num_steps = 18
   dt = 1000
[]

[Postprocessors]
  [avg_temperature]
    type = SideAverageValue
    boundary = 'left right top bottom'
    execute_on = 'initial timestep_end'
    variable = temperature
  []
  [avg_th_cond]
    type = ADElementAverageMaterialProperty
    block = 0
    mat_prop = thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [avg_th_cond_USHPRR]
    type = ADElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRR_thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [avg_th_cond_USHPRRFit]
    type = ADElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRRFit_thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [avg_specific_heat]
    type = ADElementAverageMaterialProperty
    block = 0
    mat_prop = specific_heat
    execute_on = 'initial timestep_end'
  []
  [avg_specific_heat_USHPRR]
    type = ADElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRR_specific_heat
    execute_on = 'initial timestep_end'
  []
  [avg_specific_heat_USHPRRFit]
    type = ADElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRRFit_specific_heat
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  csv = true
[]

(test/tests/thermalZr/thermal.i)

# The temperature is ramped on all faces of the cube from 300 K to 2100 K
# over 18000 s.
#
# Thermal conductivity and specific heat capacity are computed using the models
# by Fink and Leibowitz (1995) for pure zirconium.
#
# The thermal conductivity and specific heat computed by BISON for the
# temperatures shown below and compared with analytical solutions.
# The results are as follows:
#
#                 Zr k           Zr k             Zr Cp           Zr Cp
# Temp(K)     BISON(W/m-K)  analytical(W/m-K)  BISON(J/kg-K)  analytical(J/kg-K)
#  300          21.1779         21.1779          285.138         285.138
#  600          19.0001         19.0001          320.322         320.322
#  900          20.6024         20.6024          350.299         350.299
# 1200          23.4916         23.4916          307.502         307.502
# 1500          27.1693         27.1693          326.892         326.892
# 1800          31.4690         31.4690          354.447         354.447
# 2100          36.3197         36.3197          391.823         391.823

[Mesh]
  [square]
    type = GeneratedMeshGenerator
    dim = 2
  []
[]

[Variables]
  [temperature]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
  [temperatureUSHPRR]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
  [temperatureUSHPRRFit]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temperature
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temperature
  []
  [heatUSHPRR]
    type = HeatConduction
    variable = temperatureUSHPRR
  []
  [heat_ieUSHPRR]
    type = HeatConductionTimeDerivative
    variable = temperatureUSHPRR
  []
  [heatUSHPRRFit]
    type = HeatConduction
    variable = temperatureUSHPRRFit
  []
  [heat_ieUSHPRRFit]
    type = HeatConductionTimeDerivative
    variable = temperatureUSHPRRFit
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 18000'
    y = '300 2100'
  []
[]

[BCs]
  [VariableT]
    type = FunctionDirichletBC
    boundary = 'left right bottom top'
    variable = temperature
    function = temp_ramp
  []
  [VariableTUSHPRR]
    type = FunctionDirichletBC
    boundary = 'left right bottom top'
    variable = temperatureUSHPRR
    function = temp_ramp
  []
  [VariableTUSHPRRFit]
    type = FunctionDirichletBC
    boundary = 'left right bottom top'
    variable = temperatureUSHPRRFit
    function = temp_ramp
  []
[]

[Materials]
  [thermalZr]
    type = ZrThermal
    temperature = temperature
  []
  [thermalZr_USHPRR]
    type = ZrThermal
    temperature = temperature
    thermal_conductivity_model = USHPRR
    specific_heat_model = USHPRR
    base_name = USHPRR
  []
  [thermalZr_USHPRRFit]
    type = ZrThermal
    temperature = temperature
    thermal_conductivity_model = USHPRRFit
    specific_heat_model = USHPRRFit
    base_name = USHPRRFit
  []
  [density]
    type = ParsedMaterial
    block = 0
    property_name = density
    expression = 5800.0
  []
[]

[Executioner]
   type = Transient
   solve_type = PJFNK

   petsc_options = '-snes_ksp_ew'
   petsc_options_iname = '-ksp_gmres_restart'
   petsc_options_value = '101'

   line_search = 'none'

   l_max_its = 100
   nl_max_its = 100
   nl_rel_tol = 1e-8
   nl_abs_tol = 1e-10
   l_tol = 1e-5
   start_time = 0.0
   num_steps = 18
   dt = 1000
[]

[Postprocessors]
  [avg_temperature]
    type = SideAverageValue
    boundary = 'left right top bottom'
    execute_on = 'initial timestep_end'
    variable = temperature
  []
  [avg_th_cond]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [avg_th_cond_USHPRR]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRR_thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [avg_th_cond_USHPRRFit]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRRFit_thermal_conductivity
    execute_on = 'initial timestep_end'
  []
  [avg_specific_heat]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = specific_heat
    execute_on = 'initial timestep_end'
  []
  [avg_specific_heat_USHPRR]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRR_specific_heat
    execute_on = 'initial timestep_end'
  []
  [avg_specific_heat_USHPRRFit]
    type = ElementAverageMaterialProperty
    block = 0
    mat_prop = USHPRRFit_specific_heat
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  csv = true
[]

Description
Example Input Syntax
Input Parameters
Input Files
References