UZrHThermal

Computes thermal conductivity (W/m/K) and specific heat capacity (J/kg/K) for UZrH fuel based on the volume fraction of fuel constituents.

Description

Thermal properties for the uranium-zirconium-hydrogen fuel system $var element = document.getElementById("moose-equation-8490f180-6c84-4e82-bb63-f9e7dc77d35f");katex.render("(UZrH_x)", element, {displayMode:false,throwOnError:false});$ , modeled in the material UZrHThermal, are from Evans et al. (2023).

The fuel thermal conductivity is calculated as the product of the diffusivity, density, and specific heat:

$var element = document.getElementById("moose-equation-6e5fb3ab-2de3-45e8-af4b-2905e91912fe");katex.render(" k_{th} = \\alpha \\rho c_P", element, {displayMode:true,throwOnError:false});$

Where, $var element = document.getElementById("moose-equation-136eb3b7-5d96-454e-bf1a-de4e24466476");katex.render("k_{th}", element, {displayMode:false,throwOnError:false});$ is the thermal conductivity $var element = document.getElementById("moose-equation-2585adc2-668f-41ee-bf9c-080966cd69c9");katex.render("(\\frac{W}{m-K})", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-07fde3dc-f49d-46f5-9462-904e112935f8");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is the diffusivity $var element = document.getElementById("moose-equation-1010fa3f-a4fb-4457-a87b-ff7b59a7279a");katex.render("(\\frac{m^2}{s})", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-675c847c-d6dc-499e-a07f-50ca9fdc8f79");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ is the density $var element = document.getElementById("moose-equation-e416815c-c355-4bbe-89e9-643c74e71e75");katex.render("(\\frac{kg}{m^3})", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-bd53e2f1-57e4-4c40-b090-b0286cd1eadf");katex.render("c_P", element, {displayMode:false,throwOnError:false});$ is the specific heat $var element = document.getElementById("moose-equation-d493a514-332c-43e7-bd61-d3f9209cb28e");katex.render("(\\frac{J}{kg-K})", element, {displayMode:false,throwOnError:false});$ .

Because these properties vary based on the composition of the fuel (different H/Zr Ratios, $var element = document.getElementById("moose-equation-72d89580-bbc7-4c23-98df-1f89ff2bd354");katex.render("x", element, {displayMode:false,throwOnError:false});$ , and %U) a simple mixture rule is used to calculate the thermal conductivity and specific heat over this parameter space. For the thermal conductivity, the volumetric fraction of the constituents is used in the mixture calculation as follows:

$var element = document.getElementById("moose-equation-3cd5af01-99bb-48c2-affe-6f929fa9e3e5");katex.render(" k_{th,UZrH_{x}} = k_{th,U} V_{U} + k_{th,ZrH_x} V_{ZrH_{x}}", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-9a3868d3-a30f-418e-a271-b83c1b1930fb");katex.render("k", element, {displayMode:false,throwOnError:false});$ is the thermal conductivity of the constituents and $var element = document.getElementById("moose-equation-5081c190-3665-4bc0-8fb2-a0059177e14e");katex.render("V", element, {displayMode:false,throwOnError:false});$ is the volume fraction. For this model, the volume fraction of uranium is specified as an input and the volume fraction of $var element = document.getElementById("moose-equation-288be265-f840-4b88-ae1b-6f48582068b1");katex.render("ZrH_x", element, {displayMode:false,throwOnError:false});$ is calculated as:

$var element = document.getElementById("moose-equation-4919a38a-9723-456a-b99c-06c2f05e87eb");katex.render(" V_{ZrH_{x}} = 1 - V_{U}", element, {displayMode:true,throwOnError:false});$

Seperately, the fuel heat capacity, which is applied to the fuel material, is calculated using a similar method: $var element = document.getElementById("moose-equation-84bf1bb1-096e-4543-b16b-994a31e2a466");katex.render(" c_{P,UZrH_{x}} = c_{P,U} V_{U} + c_{P,ZrH_x} V_{ZrH_{x}}", element, {displayMode:true,throwOnError:false});$

Thermal Diffusivity

This model calculates both the thermal diffusivity for the $var element = document.getElementById("moose-equation-ecc26b78-018b-455a-93ba-99a979c3beed");katex.render("ZrH_x", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-ad88a3df-1117-4c5a-a46b-49ee19ffcfbf");katex.render("U", element, {displayMode:false,throwOnError:false});$ components individually to be used in the thermal conductivity calculation.

The thermal diffusivity for $var element = document.getElementById("moose-equation-9cbfe0e9-eddb-43b1-8913-6bcc0394e736");katex.render("ZrH_x", element, {displayMode:false,throwOnError:false});$ is based on the temperature and ratio of the constituents in the material. It is calculated as follows:

$var element = document.getElementById("moose-equation-3d50f67f-a9f9-4351-9f34-671725a74130");katex.render(" \\alpha_{ZrH_x} = \\left( \\frac{67.9}{T+1.62 (2-x) \\times 10^3 - 118} - 1.16 \\times 10^{-2} \\right) 10^{-4}", element, {displayMode:true,throwOnError:false});$

Where, $var element = document.getElementById("moose-equation-140f8bce-9445-4e52-98ed-c94bfea8719c");katex.render("\\alpha_{ZrH_x}", element, {displayMode:false,throwOnError:false});$ is the thermal diffusivity of the $var element = document.getElementById("moose-equation-bbf9cf30-7d13-4b9f-aaf7-04f28e049225");katex.render("ZrH_x", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-085a4ae1-b535-492e-8d65-08f3236e3bd5");katex.render("(\\frac{m^2}{s})", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-47144246-0b59-4f85-a6c5-7f81ef022363");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature ( $var element = document.getElementById("moose-equation-4186b4f5-fcb1-42f1-9e4a-fa810eececbd");katex.render("K", element, {displayMode:false,throwOnError:false});$ ), and $var element = document.getElementById("moose-equation-638272c2-9bf1-4b9b-9d6e-46aab436c549");katex.render("x", element, {displayMode:false,throwOnError:false});$ is the H-Zr Ratio ( $var element = document.getElementById("moose-equation-20a049c6-6809-4f58-82fa-92155e1910cd");katex.render("x \\ge 1.6", element, {displayMode:false,throwOnError:false});$ )

The thermal diffusivity for uranium, $var element = document.getElementById("moose-equation-63c6520e-d0e8-4cd6-985e-878d8f49cf29");katex.render("U", element, {displayMode:false,throwOnError:false});$ , is calculated as follows: $var element = document.getElementById("moose-equation-b543d809-ee95-44c5-9642-00e060796b66");katex.render(" \\alpha_{U} = \\begin{dcases} \\begin{gather*} -6.424 T^2 \\times 10^{-12} + 7.717 T \\times 10^{-9} + 8.579 \\times 10^{-6}, && T < 942 && \\alpha - phase\\\\ 7.161 T \\times 10^{-9} + 4.853 \\times 10^{-6}, && 942 \\le T < 1049 && \\beta - phase \\\\ 8.316 T \\times 10^{-9} + 5.011 \\times 10^{-6}, && 1049 \\le T < 1405 && \\gamma - phase \\end{gather*} \\end{dcases}", element, {displayMode:true,throwOnError:false});$

Where, $var element = document.getElementById("moose-equation-a4d342e7-d61b-4d50-87ad-37701d6e8947");katex.render("\\alpha_{U}", element, {displayMode:false,throwOnError:false});$ is the thermal diffusivity of the $var element = document.getElementById("moose-equation-11dd4c51-8f1b-4089-8474-b06c868b5de7");katex.render("U", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-13152a22-43a6-4b0e-b610-21910150ad55");katex.render("(\\frac{m^2}{s})", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-5ed0ece3-551e-48e6-a93c-f1df0b736a49");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature ( $var element = document.getElementById("moose-equation-d263c1c2-13c4-462e-8f33-4a3092c5e3e4");katex.render("K", element, {displayMode:false,throwOnError:false});$ ). Although the diffusivity is provided for the three phases of the metal, only the $var element = document.getElementById("moose-equation-b6845834-0352-4d9a-8e51-3bd9b6c0b50b");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ phase is used, as a phase change model is not yet implemented to account for the spatial distribution of the uranium phases.

Density

The density of the fuel material is calculated initally using the densities from the individual components. The theoretical density of the $var element = document.getElementById("moose-equation-b14b1f1d-daf1-48f9-b62a-0cb71b8b9fb2");katex.render("U", element, {displayMode:false,throwOnError:false});$ is assumed to be 19050 $var element = document.getElementById("moose-equation-7ff89416-1484-43ef-917f-955ae785754f");katex.render("\\frac{kg}{m^3}", element, {displayMode:false,throwOnError:false});$ . The density for the $var element = document.getElementById("moose-equation-70b7f9af-48f2-4059-ae72-e84a41a41581");katex.render("ZrH_x", element, {displayMode:false,throwOnError:false});$ is calculated as: $var element = document.getElementById("moose-equation-b2957fa5-91b2-4b34-8b01-9d6fc68b2527");katex.render(" \\rho_{ZrH_x} = \\begin{gather*} (1.71 \\times 10^{-4} + 4.2 \\times 10^{-6} x)^{-1}, && x \\ge 1.6 \\end{gather*}", element, {displayMode:true,throwOnError:false});$

Density changes in the fuel itself due to dimensional changes are applied in the density model using the components from all deformation mechanisms included in the simulation.

Specific Heat Capacity

The specific heat capacity for the $var element = document.getElementById("moose-equation-4228e8d4-a69a-423d-8a00-7fe0ccb5d829");katex.render("UZrH_x", element, {displayMode:false,throwOnError:false});$ fuel is calculated using the invididual heat capacity components of the constituents. For the $var element = document.getElementById("moose-equation-9bb3f11f-2448-4509-ba25-cab224854b23");katex.render("ZrH_x", element, {displayMode:false,throwOnError:false});$ the heat capacity is calculated using the temperature, mass, and composition of the material:

$var element = document.getElementById("moose-equation-befe81e6-479d-45c6-b697-1a8baea87844");katex.render(" c_{P,ZrH_x} = \\frac{(25.02 + 4.746 x + (3.103 \\times 10^{-3} + 2.008 \\times 10^{-2} x) T - (1.943 \\times 10^5 + 6.358 \\times 10^5 x) T^{-2})} {M_{ZrH_x}}", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-91ed465d-520f-43d8-bb5e-eb51680d2a76");katex.render("c_{P,ZrH_x}", element, {displayMode:false,throwOnError:false});$ is the specific heat capacity $var element = document.getElementById("moose-equation-3761c722-1036-4e00-87fb-8fbbc4a892df");katex.render("(\\frac{J}{kg-K})", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-8da967d3-c3bd-4fef-98b3-517ce5d63d87");katex.render("x", element, {displayMode:false,throwOnError:false});$ is the hydrogen-to-zirconium ratio, $var element = document.getElementById("moose-equation-878a4f57-7d7d-4cbe-b61c-ac28652a6163");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature (K), and $var element = document.getElementById("moose-equation-59d1076f-4798-43ab-8ab2-a76deb2ef390");katex.render("M_{ZrH_x}", element, {displayMode:false,throwOnError:false});$ is the molar mass (kg) of the chemical.

The molar mass $var element = document.getElementById("moose-equation-be6ff3a0-9a05-431a-bcb2-7782d54eceef");katex.render("M_{ZrH_x}", element, {displayMode:false,throwOnError:false});$ (kg/mol) is dependant on $var element = document.getElementById("moose-equation-a715ed25-7fb3-4e02-a307-6a42e98c0f3d");katex.render("x", element, {displayMode:false,throwOnError:false});$ (the hydrogen-to-zirconium ratio), and is calculated using the following expression: $var element = document.getElementById("moose-equation-2f36c702-e641-499c-88ab-8b5e163ccd8a");katex.render(" M_{ZrH_x} = \\frac{91.224 + 1.00784 x}{10^3}", element, {displayMode:true,throwOnError:false});$

The specific heat capacity for uranium, $var element = document.getElementById("moose-equation-79fdb579-ca4d-4241-90b8-892756253aeb");katex.render("U", element, {displayMode:false,throwOnError:false});$ , is calculated as follows: $var element = document.getElementById("moose-equation-334fac9e-b801-4a76-9448-1ee2c1ea5ecf");katex.render(" c_{P,U} = \\begin{dcases} \\begin{gather*} 104.82 + 5.3686 \\times 10^{-3} T + 1.01823 \\times 10^{-4} T^2, && T < 942 && \\alpha - phase\\\\ 176.4, && 942 \\le T < 1049 && \\beta - phase \\\\ 156.8, && 1049 \\le T < 1405 && \\gamma - phase \\end{gather*} \\end{dcases}", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-bdc4d23e-4bd6-4dc3-ab64-280be1cbb4b4");katex.render("c_{P,U}", element, {displayMode:false,throwOnError:false});$ is the specific heat capacity $var element = document.getElementById("moose-equation-e76e92e5-d3ed-43ed-9360-3c005f8d8ae3");katex.render("(\\frac{J}{kg-K})", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-c002004c-ed75-415d-b099-0eeb95e48bc6");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature (K). Again, only the alpha phase is considered here; values for the various phases are provided with the intent of developing a phase change model.

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [thermal_props]
    type = UZrHThermal<<<{"description": "Computes thermal conductivity (W/m/K) and specific heat capacity (J/kg/K) for UZrH fuel based on the volume fraction of fuel constituents.", "href": "UZrHThermal.html"}>>>
    specific_heat_scale_factor<<<{"description": "Optional scaling factor applied to the overall specific heat."}>>> = 1.0
    thermal_conductivity_scale_factor<<<{"description": "Optional scaling factor applied to the overall thermal conductivity."}>>> = 1.0
    U_volume_fraction<<<{"description": "Sets the volume fraction of uranium metal in the fuel"}>>> = 0.112
    HZr_ratio<<<{"description": "Sets the hydrogen-to-zirconium ratio of the fuel"}>>> = 1.6
    temperature<<<{"description": "Coupled temperature"}>>> = temp
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
[]

Input Parameters

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

Required Parameters

HZr_ratio1.6Sets the hydrogen-to-zirconium ratio of the fuel
Default:1.6
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Sets the hydrogen-to-zirconium ratio of the fuel
U_volume_fraction0.112Sets the volume fraction of uranium metal in the fuel
Default:0.112
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Sets the volume fraction of uranium metal in the fuel
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.

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

Input Files

(test/tests/uzrh_thermal/1000.i)

References

J. A. Evans, R. T. Sweet, and D. D. Keiser. MARVEL Reactor Fuel Performance Report. Technical Report INL/RPT-22-68555, Idaho National Laboratory, March 2023.

@TECHREPORT{INLRPT-22-68555,
    author = "Evans, J. A. and Sweet, R. T. and Keiser, D. D.",
    title = "{MARVEL Reactor Fuel Performance Report}",
    year = "2023",
    number = "INL/RPT-22-68555",
    month = "March",
    day = "22",
    publisher = "INL Technical Report",
    institution = "Idaho National Laboratory"
}

(test/tests/uzrh_thermal/1000.i)

# This regression test is designed to test the thermal properties
# (specific heat capacity and thermal conductivity) of the UZrH dispersoid fuel
# for a specific uranium volume fraction and H-to-Zr ratio.
#
# This is performed through calculations of the thermal conductivity and specific
# heat and a comparison to the underlying correlations
#
# Temperature = 1000K
# U_frac = .112  ---volume fraction of Uranium metal in 30%wt U fuel
# Zr_H_ratio = 1.6  -- fraction of H to Zr atoms on the matrix
# ZrH_mol_mass = (91.224 + 1.00784 * Zr_H_ratio) / 1000 = 0.092836544  --- molecular mass of ZrH in Kg/mol
#
#  K_fuel = U_frac * U_cp * U_dens * U_diff + (1 - U_frac) ZrH_cp * ZrH_dens * ZrH_diff
#  Cp_fuel = U_frac * U_cp + (1 - U_frac) ZrH_cp
#
#  U_cp = 104.82 + 0.0053686 * temp + 0.000101823 * (temp^2) = 212.0116
#  U_dens = 19050
#  U_diff = =-6.42e-12 *(temp^2) + 7.717e-9 * temp + 8.579e-6 = 9.88E-06
#
#  ZrH_cp = (25.02 + 4.746 * Zr_H_ratio + (0.003103 + 0.02008 * Zr_H_ratio) * temp -
#           (194300 + 635800 * Zr_H_ratio) * (temp^-2)) / ZrH_mol_mass = 717.746
#  ZrH_dens = ((0.000171 + 0.0000042 * Zr_H_ratio)^ -1) = 5626.83
#  ZrH_diff = (0.00679 / (temp + 1620 * (2 - Zr_H_ratio) - 118)) - 0.00000116  = 3.278E-06
#
# K_fuel = 16.22 W/m/K
# Cp_fuel = 661.10 J/kg/K
#
# BISON CALCULATIONS
# K_fuel = 16.22297 W/m/K
# CP_fuel = 661.1034 J/kg/K
#
# Differences between the BISON predictions and exact solutions agree well.

[Mesh]
  [unit_line]
    type = GeneratedMeshGenerator
    dim = 1
    xmax = 1e-2
  []
[]

[Variables]
  [temp]
    initial_condition = 1000
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []
[]

[Materials]
  [thermal_props]
    type = UZrHThermal
    specific_heat_scale_factor = 1.0
    thermal_conductivity_scale_factor = 1.0
    U_volume_fraction = 0.112
    HZr_ratio = 1.6
    temperature = temp
    outputs = all
  []
[]

[Executioner]
  type = Transient
  num_steps = 1
[]

[Postprocessors]
  [temperature_avg]
    type = ElementAverageValue
    variable = temp
  []
  [spheat_avg]
    type = ElementAverageValue
    variable = specific_heat
  []
  [thcond_avg]
    type = ElementAverageValue
    variable = thermal_conductivity
  []
[]

[Outputs]
  csv = true
  file_base = 1000_out
[]

(test/tests/uzrh_thermal/1000.i)

# This regression test is designed to test the thermal properties
# (specific heat capacity and thermal conductivity) of the UZrH dispersoid fuel
# for a specific uranium volume fraction and H-to-Zr ratio.
#
# This is performed through calculations of the thermal conductivity and specific
# heat and a comparison to the underlying correlations
#
# Temperature = 1000K
# U_frac = .112  ---volume fraction of Uranium metal in 30%wt U fuel
# Zr_H_ratio = 1.6  -- fraction of H to Zr atoms on the matrix
# ZrH_mol_mass = (91.224 + 1.00784 * Zr_H_ratio) / 1000 = 0.092836544  --- molecular mass of ZrH in Kg/mol
#
#  K_fuel = U_frac * U_cp * U_dens * U_diff + (1 - U_frac) ZrH_cp * ZrH_dens * ZrH_diff
#  Cp_fuel = U_frac * U_cp + (1 - U_frac) ZrH_cp
#
#  U_cp = 104.82 + 0.0053686 * temp + 0.000101823 * (temp^2) = 212.0116
#  U_dens = 19050
#  U_diff = =-6.42e-12 *(temp^2) + 7.717e-9 * temp + 8.579e-6 = 9.88E-06
#
#  ZrH_cp = (25.02 + 4.746 * Zr_H_ratio + (0.003103 + 0.02008 * Zr_H_ratio) * temp -
#           (194300 + 635800 * Zr_H_ratio) * (temp^-2)) / ZrH_mol_mass = 717.746
#  ZrH_dens = ((0.000171 + 0.0000042 * Zr_H_ratio)^ -1) = 5626.83
#  ZrH_diff = (0.00679 / (temp + 1620 * (2 - Zr_H_ratio) - 118)) - 0.00000116  = 3.278E-06
#
# K_fuel = 16.22 W/m/K
# Cp_fuel = 661.10 J/kg/K
#
# BISON CALCULATIONS
# K_fuel = 16.22297 W/m/K
# CP_fuel = 661.1034 J/kg/K
#
# Differences between the BISON predictions and exact solutions agree well.

[Mesh]
  [unit_line]
    type = GeneratedMeshGenerator
    dim = 1
    xmax = 1e-2
  []
[]

[Variables]
  [temp]
    initial_condition = 1000
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []
[]

[Materials]
  [thermal_props]
    type = UZrHThermal
    specific_heat_scale_factor = 1.0
    thermal_conductivity_scale_factor = 1.0
    U_volume_fraction = 0.112
    HZr_ratio = 1.6
    temperature = temp
    outputs = all
  []
[]

[Executioner]
  type = Transient
  num_steps = 1
[]

[Postprocessors]
  [temperature_avg]
    type = ElementAverageValue
    variable = temp
  []
  [spheat_avg]
    type = ElementAverageValue
    variable = specific_heat
  []
  [thcond_avg]
    type = ElementAverageValue
    variable = thermal_conductivity
  []
[]

[Outputs]
  csv = true
  file_base = 1000_out
[]

Description
Example Input Syntax
Input Parameters
Input Files
References