MOXThermal

Computes specific heat and thermal conductivity for oxide fuel.

Description

The MOXThermal model computes specific heat and thermal conductivity for mixed oxide (MOX) fuel. A number of correlations are available. This model is also compliant for use in AD-based simulations by adding the AD prefix to the type name.

Thermal Conductivity

Three models are available to compute MOX thermal properties. For these models, thermal conductivity of unirradiated material is first defined. In general, these relationships are then multiplied by correction factors to account for effects of irradiation, burnup, MOX content, and porosity. The corrections factors used in BISON have been developed by Lucuta et al. (1996) and are recommended by Carbajo et al. (2001). Choices for MOX fuel include:

Duriez-Ronchi model, a combination of models by Duriez et al. (2000) and Ronchi et al. (1999), is recommended by Carbajo et al. (2001).
Halden model is from Lanning et al. (2005).
Amaya model is proposed by Amaya et al. (2011)

1. Duriez-Ronchi model

The Duriez-Ronchi model is recommended by Carbajo et al. (2001) and is a combination of Duriez et al. (2000) and Ronchi et al. (1999) models. In this model, thermal conductivity of unirradiated MOX is given by: $var element = document.getElementById("moose-equation-7b2f6c9a-a3b2-4c31-ae24-fcae789a3cc5");katex.render("k_0(T,x) = 1.158 \\left[ \\frac{1}{A+Ct_n} + \\frac{6400}{t_n^{5/2}}\\exp{\\left(-\\frac{16.35}{t_n} \\right)} \\right] ", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-cbad9c83-32f3-487c-814a-4a675b15eda1");katex.render("k_0", element, {displayMode:false,throwOnError:false});$ is the thermal conductivity (W/m-K), $var element = document.getElementById("moose-equation-1c255146-2c1b-48a3-b327-e8cbb7900bb3");katex.render("x", element, {displayMode:false,throwOnError:false});$ is a deviation from stoichiometry (unitless), and the following expressions define the remaining terms: $var element = document.getElementById("moose-equation-a388ead8-047e-4d42-a8bf-a18315e0a5e1");katex.render("\\begin{aligned} t_n & = T \\text{ (K)} / 1000 \\text{ (reduced temperature)}\\\\ A(x) & = 2.85 x + 0.035\\\\ C(x) & = -0.715 x +0.286 \\end{aligned}", element, {displayMode:true,throwOnError:false});$ This model provides temperature and deviation from stoechiometry. It is valid from 700 to 3100 K, $var element = document.getElementById("moose-equation-6411a589-e0b5-4f40-975f-9cca6e33e402");katex.render("x", element, {displayMode:false,throwOnError:false});$ less than 0.05, and plutonium concentration between 3 and 15 wt.%. According to Carbajo et al. (2001), thermal conductivity does not depend on Pu concentration in this range. Thus this model is valid essentially for thermal reactor MOX.

2. Halden model

The Halden correlation discussed in the previous section for uranium fuel is also applicable, with one change, to MOX fuel. Reduction in thermal conductivity due to the presence of mixed oxides is accounted for by multiplying the k $var element = document.getElementById("moose-equation-aeaa0770-630e-43c1-bc8e-b39a903a9ac8");katex.render("_\\mathrm{phonon}", element, {displayMode:false,throwOnError:false});$ term by 0.92 for UO $var element = document.getElementById("moose-equation-1f9d80f9-c7b9-465f-8fae-45593e3bb695");katex.render("_2", element, {displayMode:false,throwOnError:false});$ fuel. This is consistent with the statement above regarding the lack of dependence of MOX thermal conductivity on Pu concentration. The k $var element = document.getElementById("moose-equation-bacb0a56-45e6-4474-b3fa-fc145e5e03df");katex.render("_\\mathrm{electronic}", element, {displayMode:false,throwOnError:false});$ part of the equation and the porosity correction to account for the theoretical density (TD) of interest are unchanged.

Figure 1 is a comparison of the Fink-Lucuta (for reference) urania correlation and the Fink-Amaya, Duriez-Ronchi, and Halden correlations for MOX for unirradiated 95% TD MOX fuel with 0.07% Pu concentration.

Figure 1: Unirradiated thermal conductivities for UO $var element = document.getElementById("moose-equation-22b5e506-3afa-464b-b781-6905e42d7674");katex.render("_2", element, {displayMode:false,throwOnError:false});$ (for reference) and MOX from different models implemented in BISON. Results are for 95% TD and Pu concentraton of 7 weight percent. Correction factors appropriate for each correlation have been applied.

3. Amaya model

The model available in BISON has been proposed by Amaya et al. (2011). Unlike the previous model, Amaya provides a plutonium concentration dependence. It starts from pure UO $var element = document.getElementById("moose-equation-694d82b0-77be-459c-919a-d9b42e03169a");katex.render("_2", element, {displayMode:false,throwOnError:false});$ thermal conductivity and applies corrections to account for Pu content. Unirradiated MOX thermal conductivity is given by: $var element = document.getElementById("moose-equation-fc554b4c-3a7b-43a1-a9fe-1267c9ea1103");katex.render("\\lambda_{MOX,0} = \\sqrt{\\frac{\\lambda_0}{D_{0,Pu} \\exp (D_{1,Pu} T) y}} \\mathrm{arctan}(\\sqrt{D_{0,Pu} \\exp (D_{1,Pu} T) y \\lambda_0})", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-5a3be878-e09f-471f-a62d-b04799df1090");katex.render("\\lambda_{MOX,0}", element, {displayMode:false,throwOnError:false});$ is the MOX unirradiated thermal conductivity ( $var element = document.getElementById("moose-equation-3fc4fe4e-ffa9-42b3-8037-57357ac5fc1f");katex.render("\\mathrm{W} \\mathrm{m}^{-1} \\mathrm{K}^{-1}", element, {displayMode:false,throwOnError:false});$ ), $var element = document.getElementById("moose-equation-5984b5e1-e3c4-4c87-82cb-ffac4560a4f4");katex.render("\\lambda_0", element, {displayMode:false,throwOnError:false});$ is the UO $var element = document.getElementById("moose-equation-feb18dd9-cafa-4596-aecf-9812e6f0586b");katex.render("_2", element, {displayMode:false,throwOnError:false});$ unirradiated thermal conductivity ( $var element = document.getElementById("moose-equation-58d753e7-1d13-4e7a-8ef4-63e966c56dc0");katex.render("\\mathrm{W} \\mathrm{m}^{-1} \\mathrm{K}^{-1}", element, {displayMode:false,throwOnError:false});$ ), $var element = document.getElementById("moose-equation-0a315e8c-3d7f-4fd8-bbf1-a907693ee1ce");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature (K), $var element = document.getElementById("moose-equation-675651e0-07dd-451e-899a-024f6de60c8a");katex.render("y", element, {displayMode:false,throwOnError:false});$ is the plutonium concentration (weight percent), and the following constants are used: $var element = document.getElementById("moose-equation-dfdc3eec-cfa3-4a0c-af1e-3bec3e133e2a");katex.render("\\begin{aligned} & D_{0,Pu} = 0.209\\textrm{ m}\\textrm{W}\\textrm{K}^{-1}\\\\ & D_{1,Pu} = 1.09 10^{-3}\\textrm{ K}^{-1} \\end{aligned}", element, {displayMode:true,throwOnError:false});$

Figure 2: Unirradiated thermal conductivities for UO $var element = document.getElementById("moose-equation-b020aa75-50d6-4357-b184-00a73513375e");katex.render("_2", element, {displayMode:false,throwOnError:false});$ and MOX from different models implemented in BISON.

BISON uses Fink model to compute unirradiated UO $var element = document.getElementById("moose-equation-d96c63d0-213a-47ce-ae87-9d69a30f1d88");katex.render("_2", element, {displayMode:false,throwOnError:false});$ thermal conductivity. Amaya model's coefficients have been fitted in the temperature range from 400 K to 1500 K and the plutonium concentration up to 30 wt.% (Amaya et al., 2011). Figure 2 shows a comparison of the computed thermal conductivities for the Fink-Lucuta (for reference), Fink-Amaya, and Duriez-Ronchi models for unirradiated MOX at 95% TD.

Crumbled thermal conductivity

When simulating LOCA conditions some portions of the fuel column may collapse into a crumbled mixture of fuel fragments and gas. In these regions the effective fuel thermal conductivity of the mixture is calculated by the model proposed by Chiew and Glandt (1983). This model can be used for both UO $var element = document.getElementById("moose-equation-48f88329-7266-44d5-844e-8119350619f5");katex.render("_2", element, {displayMode:false,throwOnError:false});$ and MOX fuel.

The correlation is given by: $var element = document.getElementById("moose-equation-6e827010-243f-4b24-a521-9c380ee8dcf2");katex.render("k_{eff} = \\frac{\\left(1 - \\beta \\right)}{\\left(1 + 2\\beta \\right) \\left(1 - \\beta \\phi \\right)} \\left(1 + 2\\beta \\phi + \\left(K_2 - 3 \\beta^2\\right)\\phi^2\\right)k_f", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-bd77cc51-3a8d-4984-9ce4-d6d675e21f8b");katex.render("\\beta", element, {displayMode:false,throwOnError:false});$ is the reduced thermal polarizability, $var element = document.getElementById("moose-equation-10a1ac10-2cfd-41b6-aa90-fc217cebe19b");katex.render("k_f", element, {displayMode:false,throwOnError:false});$ is the thermal conductivity of the fuel determined by one of the models above, $var element = document.getElementById("moose-equation-1baaa074-603b-4f6e-b0c7-e4c4c4b7d450");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$ is the packing fraction of the crumbled fuel, and $var element = document.getElementById("moose-equation-43a633a9-f047-4500-9690-7dc2d4cc7bb0");katex.render("K_2", element, {displayMode:false,throwOnError:false});$ is a function of $var element = document.getElementById("moose-equation-11340a2f-d69a-4503-9612-87acbf6c2977");katex.render("\\beta", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-59908ec6-3cba-489b-b376-a451e4bdc462");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$ defined later. The reduced thermal polarizability is given by: $var element = document.getElementById("moose-equation-45af6eb0-a1f3-4874-9bf7-0dbdeea14969");katex.render("\\beta = \\frac{k_f - k_g}{k_f + 2k_g}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-953ee819-59ae-408f-91e0-2dec000feb86");katex.render("k_g", element, {displayMode:false,throwOnError:false});$ is the thermal conductivity of the gas surrounding the crumbled fuel particles. The $var element = document.getElementById("moose-equation-f08e7c49-9473-4096-9392-4c56af142e00");katex.render("K_2", element, {displayMode:false,throwOnError:false});$ function is approximated by: $var element = document.getElementById("moose-equation-bb475f9c-79e8-46a5-b431-13e83fc78907");katex.render("K_2\\left(\\beta,\\phi\\right) = K_2^{\\left(0\\right)}\\left(\\beta \\right) + K_2^{\\left(1\\right)}\\left(\\beta \\right) \\phi", element, {displayMode:true,throwOnError:false});$ where Jernkvist and Massih (2015) used best fit approximations to the tabulated values of Chiew and Glandt (1983) to obtain: $var element = document.getElementById("moose-equation-8b5db403-be79-4842-95a4-ddb8614e316b");katex.render("K_2^{\\left(0\\right)}\\left(\\beta \\right) = 1.7383\\beta^3 + 2.8796\\beta^2 - 0.11604\\beta", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-27b51b90-c3a5-4987-9072-e28c665ceec6");katex.render("K_2^{\\left(1\\right)}\\left(\\beta \\right) = 2.8341\\beta^3 - 0.13455\\beta^2 - 0.27858\\beta", element, {displayMode:true,throwOnError:false});$

Specific Heat Capacity

The specific heat capacity (J/kg-K) is expressed by $(1)var element = document.getElementById("moose-equation-0114ea98-82fa-4d0f-a056-e0a00d0f782c");katex.render(" c_P = \\frac{ K_1 \\theta^2 \\exp\\left(\\frac{\\theta}{T}\\right) }{T^2\\left[ \\exp\\left(\\frac{\\theta}{T}\\right)-1 \\right]^2} + K_2 T + \\frac{Y K_3 E_D}{2RT^2}\\exp\\left(-\\frac{E_D}{RT}\\right)", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-b3414647-745d-4bb2-8f5c-734bd3f11f34");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature (K), $var element = document.getElementById("moose-equation-61f4556d-c140-4a27-8d95-ff584e0d260a");katex.render("Y", element, {displayMode:false,throwOnError:false});$ is the oxygen-to-metal ratio (dimensionless), and $var element = document.getElementById("moose-equation-1f3854b5-6050-4b8c-bd3a-9da1f894d10b");katex.render("R", element, {displayMode:false,throwOnError:false});$ is the universal gas constant (8.3145 J/mol-K). The empirical coefficients ( $var element = document.getElementById("moose-equation-89275e6c-278c-4792-9c0a-d55c44f218c1");katex.render("K_1", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-aada8359-4f2e-4278-a6e4-c5bbbd826195");katex.render("K_2", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-8582097c-9238-4b43-808a-11bca7ed84a3");katex.render("K_3", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-885884dc-20e5-4802-9162-f88276da4b5b");katex.render("\\theta", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-09385d4c-c873-4b12-b61e-43c451af808f");katex.render("E_D", element, {displayMode:false,throwOnError:false});$ ) are tabulated in Table 1 for UO $var element = document.getElementById("moose-equation-e590a5a8-feba-4699-8a5a-1989af67cfe6");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , and PuO $var element = document.getElementById("moose-equation-0a7a2bc9-ac59-47a3-b597-6d4e45d3a121");katex.render("_2", element, {displayMode:false,throwOnError:false});$ .

The specific heat capacity of MOX fuel is computed as: $var element = document.getElementById("moose-equation-46f60033-ff11-4cb4-a643-9f655887386b");katex.render("c_P = (1-w) c_{P,UO_2} + w c_{P,PuO_2},", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-0d0147de-a515-4dd4-aa17-cc5718d217b3");katex.render("w", element, {displayMode:false,throwOnError:false});$ is the fraction of gadolina or plutonium content (dimensionless) and $var element = document.getElementById("moose-equation-6480684a-ec12-4812-bd3d-baa332db6cd0");katex.render("c_{P,i}", element, {displayMode:false,throwOnError:false});$ is the specific heat capacity for $var element = document.getElementById("moose-equation-bcfa1d79-8d97-4762-a5fc-4ca7b3f4aef1");katex.render("i=", element, {displayMode:false,throwOnError:false});$ UO $var element = document.getElementById("moose-equation-5cd8cb9c-568d-4af2-9c23-0ab8f52f0f9d");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , and PuO $var element = document.getElementById("moose-equation-33cccaf2-6255-4198-a565-3ff951b2e66c");katex.render("_2", element, {displayMode:false,throwOnError:false});$ .

Table 1: The empirical constants used in Eq. (1) from Luscher et al. (2015).

Constant	$var element = document.getElementById("moose-equation-8eea0f33-0a24-4c16-8515-fde3b6bcdce3");katex.render("K_1", element, {displayMode:false,throwOnError:false});$ (J/kg-K)	$var element = document.getElementById("moose-equation-192914ec-9341-41e5-a838-942f3fbabdec");katex.render("K_2", element, {displayMode:false,throwOnError:false});$ (J/kg-K $var element = document.getElementById("moose-equation-a534bfdf-6513-422b-97f2-5bdb51cb46db");katex.render("^2", element, {displayMode:false,throwOnError:false});$ )	$var element = document.getElementById("moose-equation-baca2170-8247-4c59-b65e-9474fd4d9e43");katex.render("K_3", element, {displayMode:false,throwOnError:false});$ (J/kg)	$var element = document.getElementById("moose-equation-c7acd57b-a626-443d-87ee-43206c45d2ba");katex.render("\\theta", element, {displayMode:false,throwOnError:false});$ (K)	$var element = document.getElementById("moose-equation-c1504ea6-e698-456a-aabd-b5590c5373a5");katex.render("E_D", element, {displayMode:false,throwOnError:false});$ (J/mol)
UO $var element = document.getElementById("moose-equation-ba957ed8-5cd3-4d5f-bb68-68045c32311b");katex.render("_2", element, {displayMode:false,throwOnError:false});$	296.7	2.43 $var element = document.getElementById("moose-equation-e75d0b6c-a02e-4611-85c3-419584a6e4b6");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-f8829438-0a3f-47d5-b906-a77d8ea00095");katex.render("^{-2}", element, {displayMode:false,throwOnError:false});$	8.745 $var element = document.getElementById("moose-equation-fd1aaffd-5422-4372-a630-66989e5802e7");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-bf0c34e1-b85b-4df3-8e03-bf9195bb48ad");katex.render("^7", element, {displayMode:false,throwOnError:false});$	535.285	1.577 $var element = document.getElementById("moose-equation-bae2dba1-a72c-4315-bdda-c8b93cd5c252");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-c87915ee-06c2-47f8-be2d-c3ce2cf33fe2");katex.render("^5", element, {displayMode:false,throwOnError:false});$
PuO $var element = document.getElementById("moose-equation-39c7086e-e6a8-4650-abab-68f04570c802");katex.render("_2", element, {displayMode:false,throwOnError:false});$	347.4	3.95 $var element = document.getElementById("moose-equation-bae40a2b-3914-4290-a95e-57c8c135aa11");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-fb3c9ede-ad6b-4412-8777-0724341711cf");katex.render("^{-4}", element, {displayMode:false,throwOnError:false});$	3.860 $var element = document.getElementById("moose-equation-2ddaff36-9c1b-487d-85b9-896043ecf207");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-99a506c7-71fb-4973-b27c-cfde2712f9fc");katex.render("^7", element, {displayMode:false,throwOnError:false});$	571.0	1.967 $var element = document.getElementById("moose-equation-1a0ef4a0-d755-4387-8de2-489bf7ee0dea");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-7b4486fe-fbc1-4f41-8bfa-61008dd9ae80");katex.render("^5", element, {displayMode:false,throwOnError:false});$

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [fuel_thermal]
    type = MOXThermal<<<{"description": "Computes specific heat and thermal conductivity for oxide fuel.", "href": "MOXThermal.html"}>>>
    temperature<<<{"description": "Coupled temperature"}>>> = T
    burnup<<<{"description": "Coupled burnup"}>>> = burnup
    thermal_conductivity_model<<<{"description": "The thermal conductivity model."}>>> = HALDEN
    oxy_to_metal_ratio<<<{"description": "Oxygen-to-metal ratio."}>>> = 2.0
    initial_porosity<<<{"description": "Initial porosity.  Must be between 0.0 and 1.0."}>>> = 0.05
    Pu_content<<<{"description": "Weight fraction of Pu in MOX fuel (typically ~0.07)."}>>> = 0.07
  []
[]

Input Parameters

temperatureCoupled temperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled temperature
thermal_conductivity_modelThe thermal conductivity model.
C++ Type:MooseEnum
Options:DURIEZ, AMAYA, HALDEN
Controllable:No
Description:The thermal conductivity model.

Required Parameters

Pu_content0Weight fraction of Pu in MOX fuel (typically ~0.07).
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Weight fraction of Pu in MOX fuel (typically ~0.07).
axial_relocation_objectName of the AxialRelocationUserObject that determines whether the fuel has crumbled.
C++ Type:UserObjectName
Controllable:No
Description:Name of the AxialRelocationUserObject that determines whether the fuel has crumbled.
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
burnupCoupled burnup
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled burnup
burnup_functionBurnup function
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Burnup function
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.
gap_thermal_conductivityThe layered average thermal conductivity across the gas gap.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The layered average thermal conductivity across the gas gap.
initial_porosity0.05Initial porosity. Must be between 0.0 and 1.0.
Default:0.05
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Initial porosity. Must be between 0.0 and 1.0.
oxy_to_metal_ratio2Oxygen-to-metal ratio.
Default:2
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Oxygen-to-metal ratio.
porosityCoupled porosity
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled porosity
porosity_materialFalseWhether a material property for porosity is supplied.
Default:False
C++ Type:bool
Controllable:No
Description:Whether a material property for porosity is supplied.

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/ThermalFuel_error_messages/Fink-Amaya_error.i)
(test/tests/thermalMOX/ThermalMOX_porosity_function_test.i)
(test/tests/mox_thermal/Duriez/ad_test.i)
(test/tests/mox_thermal/Duriez/test.i)
(test/tests/mox_thermal/Halden/ad_test.i)
(test/tests/mox_thermal/Halden/test.i)
(test/tests/thermalMOX/ThermalMOX_test.i)
(test/tests/mox_thermal/Halden/negative_temperature_exception.i)
(test/tests/mox_thermal/Amaya/test.i)
(test/tests/mox_thermal/Amaya/ad_test.i)

References

M. Amaya, J. Nakamura, F. Nagase, and T. Fuketa. Thermal conductivity evaluation of high burnup mixed-oxide (mox) fuel pellet. Journal of Nuclear Materials, 414:303–308, 2011.

@ARTICLE{AmayaJNM2011,
    author = "Amaya, M. and Nakamura, J. and Nagase, F. and Fuketa, T.",
    title = "Thermal conductivity evaluation of high burnup mixed-oxide (MOX) fuel pellet",
    journal = "Journal of Nuclear Materials",
    year = "2011",
    volume = "414",
    pages = "303-308"
}

J. Carbajo, L. Gradyon, S. Popov, and V. Ivanov. A review of the thermophysical properties of mox and uo2 fuels. Journal of Nuclear Materials, 299:181–198, 2001.

@ARTICLE{CarbajoJNM2001,
    author = "Carbajo, J. and Gradyon, L. and Popov, S. and Ivanov, V.",
    title = "A review of the thermophysical properties of MOX and UO2 fuels",
    journal = "Journal of Nuclear Materials",
    year = "2001",
    volume = "299",
    pages = "181-198"
}

Y. C. Chiew and E. D. Glandt. The effect of structure on the conductivity of a dispersion. Journal of Colloid and Interface Science, 91(1):90–104, 1983. doi:10.1016/0021-9797(83)90238-2.

@article{chiew1983,
    author = "Chiew, Y. C. and Glandt, E. D.",
    title = "The effect of structure on the conductivity of a dispersion",
    journal = "Journal of Colloid and Interface Science",
    volume = "91",
    number = "1",
    year = "1983",
    doi = "10.1016/0021-9797(83)90238-2",
    pages = "90-104"
}

C. Duriez, J.-P. Alessandri, T. Gervais, and Y. Philipponneau. Thermal conductivity of hypostoichiometriclow pu content mixed oxide. Journal of Nuclear Materials, 277:143–158, 2000.

@ARTICLE{DuriezJNM2000,
    author = "Duriez, C. and Alessandri, J.-P. and Gervais, T. and Philipponneau, Y.",
    title = "Thermal conductivity of hypostoichiometriclow Pu content mixed oxide",
    journal = "Journal of Nuclear Materials",
    year = "2000",
    volume = "277",
    pages = "143-158"
}

L. O. Jernkvist and A. Massih. Model for axial relocation of fragmented and pulverized fuel pellets in distending fuel rods and its effects on fuel rod heat load. Technical Report SSM-2015:37, Strål säkerhets myndigheten, 2015.

@TECHREPORT{jernkvist2015,
    author = "Jernkvist, L. O. and Massih, A.",
    title = "Model for axial relocation of fragmented and pulverized fuel pellets in distending fuel rods and its effects on fuel rod heat load",
    year = "2015",
    number = "SSM-2015:37",
    institution = {Str\r{a}l s\"{a}kerhets myndigheten}
}

D. D. Lanning, C. E. Beyer, and K. J. Geelhood. Frapcon-3 updates, including mixed-oxide fuel properties. Technical Report NUREG/CR-6534, Vol. 4 PNNL-11513, Pacific Northwest National Laboratory, 2005.

@TECHREPORT{lanning2005,
    author = "Lanning, D. D. and Beyer, C. E. and Geelhood, K. J.",
    title = "FRAPCON-3 Updates, Including Mixed-Oxide Fuel Properties",
    year = "2005",
    number = "NUREG/CR-6534, Vol. 4 PNNL-11513",
    pages = "3.4",
    publisher = "Pacific Northwest National Laboratory",
    institution = "Pacific Northwest National Laboratory"
}

P.G. Lucuta, Hj. Matzke, and I.J. Hastings. A pragmatic approach to modelling thermal conductivity of irradiated UO$_2$ fuel: review and recommendations. Journal of Nuclear Materials, 232(2-3):166–180, 1996. URL: http://www.sciencedirect.com/science/article/pii/S0022311596004047, doi:10.1016/S0022-3115(96)00404-7.

@ARTICLE{Lucuta1996,
    author = "Lucuta, P.G. and Matzke, Hj. and Hastings, I.J.",
    title = "A pragmatic approach to modelling thermal conductivity of irradiated {UO}$\_2$ fuel: Review and recommendations",
    journal = "Journal of Nuclear Materials",
    year = "1996",
    volume = "232",
    pages = "166-180",
    number = "2-3",
    bdsk-url-1 = "http://www.sciencedirect.com/science/article/pii/S0022311596004047",
    bdsk-url-2 = "http://dx.doi.org/10.1016/S0022-3115(96)00404-7",
    doi = "10.1016/S0022-3115(96)00404-7",
    issn = "0022-3115",
    url = "http://www.sciencedirect.com/science/article/pii/S0022311596004047"
}

WJ Luscher, KJ Geelhood, and IE Porter. Material property correlations: comparisons between FRAPCON-4.0, FRAPTRAN-2.0, and MATPRO. Technical Report PNNL-19417 Rev. 2, Pacific Northwest National Laboratory, 9 2015.

@TECHREPORT{luscher2015,
    author = "Luscher, WJ and Geelhood, KJ and Porter, IE",
    title = "Material Property Correlations: Comparisons between {FRAPCON-4.0}, {FRAPTRAN-2.0}, and {MATPRO}",
    year = "2015",
    month = "9",
    institution = "Pacific Northwest National Laboratory",
    number = "PNNL-19417 Rev. 2"
}

C. Ronchi, M. Sheindlin, M. Musella, and G.J. Hyland. Thermal conductivity of uranium dioxide up to 2900 K from simultaneous measurement of the heat capacity and thermal diffusivity. Journal of Applied Physics, 85:776–789, 1999.

@ARTICLE{RonchiJAP1999,
    author = "Ronchi, C. and Sheindlin, M. and Musella, M. and Hyland, G.J.",
    title = "Thermal conductivity of uranium dioxide up to 2900 {K} from simultaneous measurement of the heat capacity and thermal diffusivity",
    journal = "Journal of Applied Physics",
    year = "1999",
    volume = "85",
    pages = "776-789"
}

(test/tests/mox_thermal/Halden/test.i)

# This test case is prepared to test the thermal conductivity using the Halden model.
#
# The temperature is ramped on all BCs of the unit line from 500 K to 3000K over 100 Ms.
# The fission rate is 2e19 n/m3/s, so that the burnup goes from 0 to
#       ~ 8.6 at.% (82 MWd/kgU) at the end of the simulation.
#
# Thermal conductivity of UO2 fuel at 95%TD is computed using the Halden UO2 thermal conductivity model.
# The BISON predictions (BISON_k) compared to the expected results (expected_k)
# are summarized (see: post_processing.py --> comparison_results.csv) as:
#
# expected_k       BISON_k    percent_error  expected_Pu_k  BISON_Pu_k  percent_error_Pu
# 5.869784658    5.869784658    -7.55E-13    5.401819451    5.401819451   -1.55E-13
# 5.580850701    5.580850701     5.44E-13    5.136078051    5.136078051   -3.55E-13
# 5.320173923    5.320173923    -5.55E-13    4.896337002    4.896337002    6.55E-13
# 5.083833746    5.083833746     3.11E-13    4.678989552    4.678989552    2.11E-13
# 4.86860483     4.86860483      1.44E-13    4.481068576    4.481068576   -5.77E-13

[GlobalParams]
  order = FIRST
  family = LAGRANGE
[]

[Mesh]
  [geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type= EDGE2
  []
[]

[Variables]
  [T]
    initial_condition = 500
  []
[]

[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
  [th_cond]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = T
  []
  [heat_source]
     type = NeutronHeatSource
     variable = T
     energy_per_fission = 3.2e-11
     fission_rate = fission_rate
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable= fission_rate
    value = 2e19 # Standard fission_rate fissions/m^3/s
    execute_on = 'initial timestep_begin'
  []
  [th_cond]
    type = MaterialRealAux
    variable = th_cond
    property = thermal_conductivity
    execute_on = 'initial linear'
  []
  [burnup]
    type = BurnupAux
    variable = burnup
    density = 10431.0 # 95% TD
    fission_rate = fission_rate
    execute_on = timestep_begin
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 100.e6'
    y = '500 3000'
  []
[]

[BCs]
   [VariableT]
     type = FunctionDirichletBC
     boundary = 'left right'
     variable = T
     function = temp_ramp
   []
[]

[Materials]
  [fuel_thermal]
    type = MOXThermal
    temperature = T
    burnup = burnup
    thermal_conductivity_model = HALDEN
    oxy_to_metal_ratio = 2.0
    initial_porosity = 0.05
    Pu_content = 0.07
  []
  [density]
    type = ParsedMaterial
    property_name = density
    expression = 10431.0
  []
[]

[Executioner]
  type = Transient
  start_time = 0
  num_steps = 100
  dt = 1.0e6
[]

[Postprocessors]
   [avg_th_cond]
     type = ElementAverageValue
     variable = th_cond
     execute_on = 'initial timestep_end'
   []
   [average_rod_burnup]
     type = ElementAverageValue
     variable = burnup
     execute_on = timestep_end
   []
   [average_fuel_T]
     type = ElementAverageValue
     variable = T
     execute_on = 'initial timestep_end'
   []
 []

[Outputs]
  csv = true
[]

(test/tests/ThermalFuel_error_messages/Fink-Amaya_error.i)

#Error testing when Pu_content = 0 in Fink-Amaya model.
[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
    xmax = 0.001
    ymax = 0.001
  []
[]

[Variables]
  [temp]
    initial_condition = 200
  []
[]
[AuxVariables]
  [burnup]
  []
[]

[Kernels]
  [heat_conduction]
    type = HeatConduction
    variable = temp
  []
  [heat_source]
    type = HeatSource
    variable = temp
    value = 1e10
  []
[]

[AuxKernels]
  [burnup]
    type = ConstantAux
    variable = burnup
    value = 0.01
  []
[]

[Materials]
  [heat_conductor]
    type = MOXThermal
    block = 0
    thermal_conductivity_model = AMAYA
    Pu_content = 0
    temperature = temp
    burnup = burnup
    initial_porosity = 0.001
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = temp
    boundary = 'left right'
    value = 200
  []
[]

[Executioner]
  type = Steady

  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = false
[]

(test/tests/thermalMOX/ThermalMOX_porosity_function_test.i)

# The mesh is 2 1x1x1 cubes. The mesh density is of 1 interval per edge.
# The temperature is ramped on all faces of each cube from 500 K to 2000K,
# then back to 1500.
#
# Porosity is now defined using PorosityMOX, where porosity is initially
# 0.05 at 500K then decreases linearly to 0 at 1973K.
#
# The purpose of PorosityMOX is for porosity to be defined as a function of
# temperature, or a constant.
# Additionally, in the case when porosity is a funciton of temperature an
# annealing temperature is defined.
# When the annealing temperature is reached, the porosity remains constant
# (whatever the value of porosity is at the annealing temperature)
# regardless of subsequent values of temperature.

# For this test problem, the porosity should stay 0 after the annealing temperature
# is reached (1973K in this case).
#
# The fission rate is from 2e19 n/m3/s, so that the burnup is from 5 at.%
# at the end of the simulation.
# Unirradiated thermal conductivity of block 1 is calculated using Duriez
# model. Then Lucuta's corrections are applied to account for burnup effect.
# Unirradiated thermal conductivity of block 2 is calculated using Fink-Amaya
# model. Then Lucuta's corrections for burnup are applied. For block2, a Pu
# content of 7 wt.% is assumed in MOX.
#
#
# The thermal conductivity computed by BISON was picked up each 10 time
# steps for each block, and compared with analytical solution for both Duriez
# and Amaya's models.The results are the following:
# DURIEZ
# Temp (k)   Bu (at. %)    Porosity    BISON k (W/m/K)   analytical k (W/m/K)
# 700.172     0.999619      0.043205    3.39382           3.39383
# 900.344     1.99924       0.036411    2.81614           2.81614
# 1100.52     2.99886       0.029616    2.54114           2.54113
# 1300.69     3.99848       0.022821    2.25040           2.25039
# 1500.90     4.99810       0.016027    2.04159           2.04155
# 1700.70     5.99770       0.009244    1.92175           1.92174
# 1900.50     6.99730       0.002460    1.89503           1.89503
# 1999.60     7.49710       0.0         1.91083           1.91084
# 1898.60     7.99700       0.0         1.87649           1.87649
# 1696.70     8.99660       0.0         1.86866           1.86866
# 1500.00     9.99620       0.0         1.92874           1.92873
# AMAYA
# Temp (k)   Bu (at. %)    Porosity    BISON k (W/m/K)   analytical k (W/m/K)
# 700.172     0.999619      0.043205    3.40254           3.40255
# 900.344     1.99924       0.036411    2.86709           2.86710
# 1100.52     2.99886       0.029616    2.58204           2.58203
# 1300.69     3.99848       0.022821    2.25723           2.25723
# 1500.90     4.99810       0.016027    2.00470           2.00466
# 1700.70     5.99770       0.009244    1.83369           1.83368
# 1900.50     6.99730       0.002460    1.74515           1.74516
# 1999.60     7.49710       0.0         1.72517           1.72517
# 1898.60     7.99700       0.0         1.72872           1.72872
# 1696.70     8.99660       0.0         1.78418           1.78418
# 1500.00     9.99620       0.0         1.89409           1.89409

initial_fuel_density = 10431.0

[Mesh]
  [mesh]
    type = FileMeshGenerator
    file = 2cubes.e
  []
[]

[Variables]
  [T]
    order = FIRST
    family = LAGRANGE
    initial_condition = 500     # set initial T to 500 K
  []
[]

[AuxVariables]
  [porosity_var]
    order = CONSTANT
    family = MONOMIAL
    block = '1 2'
  []
  [fission_rate]
    order = FIRST
    family = LAGRANGE
    block = '1 2'
  []
  [burnup]
    order = FIRST
    family = LAGRANGE
    block = '1 2'
  []
  [th_cond]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = T
  []
  [heat_source]
     type = NeutronHeatSource
     variable = T
     block = '1 2'
     energy_per_fission = 3.2e-11
     fission_rate = fission_rate
  []
[]

[AuxKernels]
  [porosity_aux]
    type = MaterialRealAux
    variable = porosity_var
    property = porosity
  []
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable= fission_rate
    block = '1 2'
    value = 2e19     # Standard fission_rate
    execute_on = 'initial timestep_begin'
  []
  [th_cond]
    type = MaterialRealAux
    variable = th_cond
    property = thermal_conductivity
    block = '1 2'
  []
  [burnup]
    type = BurnupAux
    block = '1 2'
    variable = burnup
    density = ${initial_fuel_density}
    fission_rate = fission_rate
    execute_on = timestep_begin
  []
[]

[Functions]
  [p_func]
    type = PiecewiseLinear
    x = '500  1973'
    y = '0.05 0.0'
  []
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 5.81e7 7e7  8.72e7 11.6e7'
    y = '500 1500   1000 2000   1500'
  []
[]

[BCs]
  [VariableT]
     type = FunctionDirichletBC
     boundary = 3
     variable = T
     function = temp_ramp
  []
[]

[Materials]
  [porosity]
    type = PorosityMOX
    block = '1 2'
    temperature = T
    porosity_temperature_function = p_func
#    anneal_temp = 1973
  []
  [fuel_thermal_Duriez]
    type = MOXThermal
    thermal_conductivity_model = DURIEZ
    block = 1
    temperature = T
    porosity_material = true
    burnup = burnup
#    initial_porosity = 0.05
    oxy_to_metal_ratio = 2.0
  []
  [fuel_thermal_Amaya]
    type = MOXThermal
    thermal_conductivity_model = AMAYA
    block = 2
    temperature = T
    porosity_material = true
    burnup = burnup
#    initial_porosity = 0.05
    Pu_content = 0.07
  []
  [density]
    type = ParsedMaterial
    block = '1 2'
    property_name = density
    expression = ${initial_fuel_density}
  []
[]

[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-4
   nl_abs_tol = 1e-6
   l_tol = 1e-5
   start_time = 0.0
   num_steps = 100
   dt = 1.163e6
[]

[Outputs]
  file_base = out_porosity_function
  [exodus]
    type = Exodus
  []
[]

(test/tests/mox_thermal/Duriez/ad_test.i)

# This test case is prepared to test the thermal conductivity using the Duriez MOX model.
#
# The temperature is ramped on all BCs of the unit line from 500 K to 3000K over 100 Ms.
# The fission rate is 2e19 n/m3/s, so that the burnup goes from 0 to
#       ~ 8.6 at.% (82 MWd/kgU) at the end of the simulation.
#
# Thermal conductivity of MOX fuel at 95%TD with a 7% Pu content is computed
# using the Duriez MOX thermal conductivity model.
# The BISON predictions (BISON_k) compared to the expected results (expected_k)
# are summarized (see: post_processing.py --> comparison_results.csv) as:
#
# expected_k     BISON_k       percent_error
# 4.502311344    4.502311344    -2.22E-14
# 4.331074842    4.331074842     5.77E-13
# 4.17347041     4.17347041     -2.66E-13
# 4.028219101    4.028219101    -4.22E-13
# 3.893949587    3.893949587    -1.24E-12

[GlobalParams]
  order = FIRST
  family = LAGRANGE
[]

[Mesh]
  [geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type= EDGE2
  []
[]

[Variables]
  [T]
    initial_condition = 500
  []
[]

[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
  [th_cond]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = ADHeatConduction
    variable = T
  []
  [heat_ie]
    type = ADHeatConductionTimeDerivative
    variable = T
  []
  [heat_source]
     type = ADNeutronHeatSource
     variable = T
     energy_per_fission = 3.2e-11
     fission_rate = fission_rate
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable= fission_rate
    value = 2e19 # Standard fission_rate fissions/m^3/s
    execute_on = 'initial timestep_begin'
  []
  [th_cond]
    type = ADMaterialRealAux
    variable = th_cond
    property = thermal_conductivity
    execute_on = 'initial linear'
  []
  [burnup]
    type = ADBurnupAux
    variable = burnup
    density = 10431.0 # 95% TD
    fission_rate = fission_rate
    execute_on = timestep_begin
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 100.e6'
    y = '500 3000'
  []
[]

[BCs]
   [VariableT]
     type = ADFunctionDirichletBC
     boundary = 'left right'
     variable = T
     function = temp_ramp
   []
[]

[Materials]
  [fuel_thermal]
    type = ADMOXThermal
    temperature = T
    burnup = burnup
    thermal_conductivity_model = DURIEZ
    oxy_to_metal_ratio = 2.0
    initial_porosity = 0.05
    Pu_content = 0.07
  []
  [density]
    type = ADParsedMaterial
    property_name = density
    expression = 10431.0
  []
[]

[Executioner]
  type = Transient
  start_time = 0
  num_steps = 100
  dt = 1.0e6
[]

[Postprocessors]
   [avg_th_cond]
     type = ElementAverageValue
     variable = th_cond
     execute_on = 'initial timestep_end'
   []
   [average_rod_burnup]
     type = ElementAverageValue
     variable = burnup
     execute_on = timestep_end
   []
   [average_fuel_T]
     type = ElementAverageValue
     variable = T
     execute_on = 'initial timestep_end'
   []
 []

[Outputs]
  csv = true
[]

(test/tests/mox_thermal/Duriez/test.i)

# This test case is prepared to test the thermal conductivity using the Duriez MOX model.
#
# The temperature is ramped on all BCs of the unit line from 500 K to 3000K over 100 Ms.
# The fission rate is 2e19 n/m3/s, so that the burnup goes from 0 to
#       ~ 8.6 at.% (82 MWd/kgU) at the end of the simulation.
#
# Thermal conductivity of MOX fuel at 95%TD with a 7% Pu content is computed
# using the Duriez MOX thermal conductivity model.
# The BISON predictions (BISON_k) compared to the expected results (expected_k)
# are summarized (see: post_processing.py --> comparison_results.csv) as:
#
# expected_k     BISON_k       percent_error
# 4.502311344    4.502311344    -2.22E-14
# 4.331074842    4.331074842    -1.04E-12
# 4.173470398    4.173470398     9.33E-13
# 4.028218943    4.028218943     7.11E-13
# 3.893948687    3.893948687     9.55E-13

[GlobalParams]
  order = FIRST
  family = LAGRANGE
[]

[Mesh]
  [geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type= EDGE2
  []
[]

[Variables]
  [T]
    initial_condition = 500
  []
[]

[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
  [th_cond]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = T
  []
  [heat_source]
     type = NeutronHeatSource
     variable = T
     energy_per_fission = 3.2e-11
     fission_rate = fission_rate
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable= fission_rate
    value = 2e19 # Standard fission_rate fissions/m^3/s
    execute_on = 'initial timestep_begin'
  []
  [th_cond]
    type = MaterialRealAux
    variable = th_cond
    property = thermal_conductivity
    execute_on = 'initial linear'
  []
  [burnup]
    type = BurnupAux
    variable = burnup
    density = 10431.0 # 95% TD
    fission_rate = fission_rate
    execute_on = timestep_begin
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 100.e6'
    y = '500 3000'
  []
[]

[BCs]
   [VariableT]
     type = FunctionDirichletBC
     boundary = 'left right'
     variable = T
     function = temp_ramp
   []
[]

[Materials]
  [fuel_thermal]
    type = MOXThermal
    temperature = T
    burnup = burnup
    thermal_conductivity_model = DURIEZ
    oxy_to_metal_ratio = 2.0
    initial_porosity = 0.05
    Pu_content = 0.07
  []
  [density]
    type = ParsedMaterial
    property_name = density
    expression = 10431.0
  []
[]

[Executioner]
  type = Transient
  start_time = 0
  num_steps = 100
  dt = 1.0e6
[]

[Postprocessors]
   [avg_th_cond]
     type = ElementAverageValue
     variable = th_cond
     execute_on = 'initial timestep_end'
   []
   [average_rod_burnup]
     type = ElementAverageValue
     variable = burnup
     execute_on = timestep_end
   []
   [average_fuel_T]
     type = ElementAverageValue
     variable = T
     execute_on = 'initial timestep_end'
   []
 []

[Outputs]
  csv = true
[]

(test/tests/mox_thermal/Halden/ad_test.i)

# This test case is prepared to test the thermal conductivity using the Halden model.
#
# The temperature is ramped on all BCs of the unit line from 500 K to 3000K over 100 Ms.
# The fission rate is 2e19 n/m3/s, so that the burnup goes from 0 to
#       ~ 8.6 at.% (82 MWd/kgU) at the end of the simulation.
#
# Thermal conductivity of UO2 fuel at 95%TD is computed using the Halden UO2 thermal conductivity model.
# The BISON predictions (BISON_k) compared to the expected results (expected_k)
# are summarized (see: post_processing.py --> comparison_results.csv) as:
#
# expected_k       BISON_k    percent_error  expected_Pu_k  BISON_Pu_k  percent_error_Pu
# 5.869784658    5.869784658    -7.55E-13    5.401819451    5.401819451   -1.55E-13
# 5.580864694    5.580864694     5.22E-13    5.136090924    5.136090924    3.77E-13
# 5.320199198    5.320199198    -7.11E-13    4.896360254    4.896360254    9.77E-13
# 5.083868147    5.083868147    -6.88E-13    4.6790212      4.6790212     -1.04E-12
# 4.868646629    4.868646629     2.00E-13    4.481107031    4.481107031    5.33E-13

[GlobalParams]
  order = FIRST
  family = LAGRANGE
[]

[Mesh]
  [geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type= EDGE2
  []
[]

[Variables]
  [T]
    initial_condition = 500
  []
[]

[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
  [th_cond]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = ADHeatConduction
    variable = T
  []
  [heat_ie]
    type = ADHeatConductionTimeDerivative
    variable = T
  []
  [heat_source]
     type = ADNeutronHeatSource
     variable = T
     energy_per_fission = 3.2e-11
     fission_rate = fission_rate
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable= fission_rate
    value = 2e19 # Standard fission_rate fissions/m^3/s
    execute_on = 'initial timestep_begin'
  []
  [th_cond]
    type = ADMaterialRealAux
    variable = th_cond
    property = thermal_conductivity
    execute_on = 'initial linear'
  []
  [burnup]
    type = ADBurnupAux
    variable = burnup
    density = 10431.0 # 95% TD
    fission_rate = fission_rate
    execute_on = timestep_begin
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 100.e6'
    y = '500 3000'
  []
[]

[BCs]
   [VariableT]
     type = ADFunctionDirichletBC
     boundary = 'left right'
     variable = T
     function = temp_ramp
   []
[]

[Materials]
  [fuel_thermal]
    type = ADMOXThermal
    temperature = T
    burnup = burnup
    thermal_conductivity_model = HALDEN
    oxy_to_metal_ratio = 2.0
    initial_porosity = 0.05
    Pu_content = 0.07
  []
  [density]
    type = ADParsedMaterial
    property_name = density
    expression = 10431.0
  []
[]

[Executioner]
  type = Transient
  start_time = 0
  num_steps = 100
  dt = 1.0e6
[]

[Postprocessors]
   [avg_th_cond]
     type = ElementAverageValue
     variable = th_cond
     execute_on = 'initial timestep_end'
   []
   [average_rod_burnup]
     type = ElementAverageValue
     variable = burnup
     execute_on = timestep_end
   []
   [average_fuel_T]
     type = ElementAverageValue
     variable = T
     execute_on = 'initial timestep_end'
   []
 []

[Outputs]
  csv = true
[]

(test/tests/mox_thermal/Halden/test.i)

# This test case is prepared to test the thermal conductivity using the Halden model.
#
# The temperature is ramped on all BCs of the unit line from 500 K to 3000K over 100 Ms.
# The fission rate is 2e19 n/m3/s, so that the burnup goes from 0 to
#       ~ 8.6 at.% (82 MWd/kgU) at the end of the simulation.
#
# Thermal conductivity of UO2 fuel at 95%TD is computed using the Halden UO2 thermal conductivity model.
# The BISON predictions (BISON_k) compared to the expected results (expected_k)
# are summarized (see: post_processing.py --> comparison_results.csv) as:
#
# expected_k       BISON_k    percent_error  expected_Pu_k  BISON_Pu_k  percent_error_Pu
# 5.869784658    5.869784658    -7.55E-13    5.401819451    5.401819451   -1.55E-13
# 5.580850701    5.580850701     5.44E-13    5.136078051    5.136078051   -3.55E-13
# 5.320173923    5.320173923    -5.55E-13    4.896337002    4.896337002    6.55E-13
# 5.083833746    5.083833746     3.11E-13    4.678989552    4.678989552    2.11E-13
# 4.86860483     4.86860483      1.44E-13    4.481068576    4.481068576   -5.77E-13

[GlobalParams]
  order = FIRST
  family = LAGRANGE
[]

[Mesh]
  [geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type= EDGE2
  []
[]

[Variables]
  [T]
    initial_condition = 500
  []
[]

[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
  [th_cond]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = T
  []
  [heat_source]
     type = NeutronHeatSource
     variable = T
     energy_per_fission = 3.2e-11
     fission_rate = fission_rate
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable= fission_rate
    value = 2e19 # Standard fission_rate fissions/m^3/s
    execute_on = 'initial timestep_begin'
  []
  [th_cond]
    type = MaterialRealAux
    variable = th_cond
    property = thermal_conductivity
    execute_on = 'initial linear'
  []
  [burnup]
    type = BurnupAux
    variable = burnup
    density = 10431.0 # 95% TD
    fission_rate = fission_rate
    execute_on = timestep_begin
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 100.e6'
    y = '500 3000'
  []
[]

[BCs]
   [VariableT]
     type = FunctionDirichletBC
     boundary = 'left right'
     variable = T
     function = temp_ramp
   []
[]

[Materials]
  [fuel_thermal]
    type = MOXThermal
    temperature = T
    burnup = burnup
    thermal_conductivity_model = HALDEN
    oxy_to_metal_ratio = 2.0
    initial_porosity = 0.05
    Pu_content = 0.07
  []
  [density]
    type = ParsedMaterial
    property_name = density
    expression = 10431.0
  []
[]

[Executioner]
  type = Transient
  start_time = 0
  num_steps = 100
  dt = 1.0e6
[]

[Postprocessors]
   [avg_th_cond]
     type = ElementAverageValue
     variable = th_cond
     execute_on = 'initial timestep_end'
   []
   [average_rod_burnup]
     type = ElementAverageValue
     variable = burnup
     execute_on = timestep_end
   []
   [average_fuel_T]
     type = ElementAverageValue
     variable = T
     execute_on = 'initial timestep_end'
   []
 []

[Outputs]
  csv = true
[]

(test/tests/thermalMOX/ThermalMOX_test.i)

# The mesh is 2 1x1x1 cubes. The mesh density is of 1 interval per edge.
# The temperature is ramped on all faces of each cube from 500 K to 1500K.
# The fission rate is from 2e19 n/m3/s, so that the burnup is from 5 at.%
# at the end of the simulation.
# Unirradiated thermal conductivity of block 1 is calculated using Duriez
# model. Then Lucuta's corrections are applied to account for burnup effect.
# Unirradiated thermal conductivity of block 2 is calculated using Fink-Amaya
# model. Then Lucuta's corrections for burnup are applied. For block2, a Pu
# content of 7 wt.% is assumed in MOX.
#
#
# The thermal conductivity computed by BISON was picked up each 10 time
# steps for each block, and compared with analytical solution for both Duriez
# and Amaya's models.The results are the following:
# DURIEZ
# Temp (k)   Bu (at. %)    BISON k (W/m/K)  analytical k (W/m/K)
# 700.172     0.999619      3.32809           3.32809
# 900.344     1.99924       2.70782           2.70782
# 1100.52     2.99886       2.39556           2.39556
# 1300.69     3.99848       2.07969           2.07969
# 1500.00     4.99810       1.84994           1.84994
#
# AMAYA
# Temp (K)   Bu (at. %)   BISON k (W/m/K)   analytical k (W/m/K)
# 700.172     0.999619      3.33664           3.33665
# 900.344     1.99924       2.75682           2.75682
# 1100.52     2.99886       2.43412           2.43411
# 1300.69     3.99848       2.08601           2.08601
# 1500.00     4.99810       1.81671           1.81671


# Note, you have to use PorosityMOX to define the porosity
# as a constant or a function of temperature.

initial_fuel_density = 10431.0

[Mesh]
  [mesh]
    type = FileMeshGenerator
    file = 2cubes.e
  []
[]

[Variables]
  [T]
    order = FIRST
    family = LAGRANGE
    initial_condition = 500     # set initial T to 500 K
  []
[]

[AuxVariables]
  [porosity_var]
    order = CONSTANT
    family = MONOMIAL
    block = '1 2'
  []
  [fission_rate]
    order = FIRST
    family = LAGRANGE
    block = '1 2'
  []
  [burnup]
    order = FIRST
    family = LAGRANGE
    block = '1 2'
  []
  [th_cond]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = T
  []
  [heat_source]
     type = NeutronHeatSource
     variable = T
     block = '1 2'
     energy_per_fission = 3.2e-11
     fission_rate = fission_rate
  []
[]

[AuxKernels]
  [porosity_aux]
    type = MaterialRealAux
    variable = porosity_var
    property = porosity
  []
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable= fission_rate
    block = '1 2'
    value = 2e19     # Standard fission_rate
    execute_on = 'initial timestep_begin'
  []
  [th_cond]
    type = MaterialRealAux
    variable = th_cond
    property = thermal_conductivity
    block = '1 2'
  []
  [burnup]
    type = BurnupAux
    block = '1 2'
    variable = burnup
    density = ${initial_fuel_density}
    fission_rate = fission_rate
    execute_on = timestep_begin
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 5.81e7'
    y = '500 1500'
  []
[]

[BCs]
   [VariableT]
     type = FunctionDirichletBC
     boundary = 3
     variable = T
     function = temp_ramp
   []
[]

[Materials]
  [porosity]
    type = PorosityMOX
    block = '1 2'
#    temperature = temp
    porosity = 0.05
#    anneal_temp = 200
  []
  [fuel_thermal_Duriez]
    type = MOXThermal
    thermal_conductivity_model = DURIEZ
    block = 1
    temperature = T
    burnup = burnup
#    initial_porosity = 0.05
    oxy_to_metal_ratio = 2.0
#    t_porosity = porosity
  []
  [fuel_thermal_Amaya]
    type = MOXThermal
    thermal_conductivity_model = AMAYA
    block = 2
    temperature = T
    burnup = burnup
#    initial_porosity = 0.05
    Pu_content = 0.07
#    t_porosity = porosity
  []
  [fuel_density]
    type = ParsedMaterial
    block = '1 2'
    property_name = density
    expression = ${initial_fuel_density}
  []
[]

[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-4
   nl_abs_tol = 1e-6
   l_tol = 1e-5
   start_time = 0.0
   num_steps = 50
   dt = 1.163e6
[]

[Outputs]
  file_base = out_constant
  [exodus]
    type = Exodus
  []
[]

(test/tests/mox_thermal/Halden/negative_temperature_exception.i)

# This test ensures that the moose exception to cut the timestep when a
# negative temperature is detected in MOXThermal is properly executed.

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

[Functions]
  [temperature_func]
    type = PiecewiseLinear
    x = '0 2'
    y = '300 -100.0'
  []
[]

[Variables]
  [temp]
    initial_condition = 300
  []
[]

[AuxVariables]
  [burnup]
    order = CONSTANT
    family = MONOMIAL
  []
[]

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

[AuxKernels]
  [burnup]
    type = ConstantAux
    variable = burnup
    value = 0
  []
[]

[BCs]
  [fixed_temp]
    type = FunctionDirichletBC
    variable = temp
    boundary = 'left right'
    function = temperature_func
  []
[]

[Materials]
  [thermal]
    type = MOXThermal
    thermal_conductivity_model = HALDEN
    Pu_content = 0.07
    temperature = temp
    burnup = burnup
  []
[]

[Executioner]
  type = Transient
  num_steps = 2
  dt = 2
[]

[Postprocessors]
  [dt]
    type = TimestepSize
  []
[]

[UserObjects]
  [terminator]
    type = Terminator
    expression = 'dt = 1'
    execute_on = timestep_end
  []
[]

[Outputs]
  csv = true
[]

(test/tests/mox_thermal/Amaya/test.i)

# This test case is prepared to test the thermal conductivity using the Amaya MOX model.
#
# The temperature is ramped on all BCs of the unit line from 500 K to 3000K over 100 Ms.
# The fission rate is 2e19 n/m3/s, so that the burnup goes from 0 to
#       ~ 8.6 at.% (82 MWd/kgU) at the end of the simulation.
#
# Thermal conductivity of MOX fuel at 95%TD with a 7% Pu content is computed
# using the Amaya MOX thermal conductivity model.
# The BISON predictions (BISON_k) compared to the expected results (expected_k)
# are summarized (see: post_processing.py --> comparison_results.csv) as:
#
# expected_k      BISON_k      percent_error
# 4.30774878     4.30774878      4.33E-13
# 4.178849628    4.178849628     4.00E-13
# 4.057206997    4.057206997    -2.44E-13
# 3.942475407    3.942475407    -1.13E-12
# 3.834071383    3.834071383    -4.88E-13

[GlobalParams]
  order = FIRST
  family = LAGRANGE
[]

[Mesh]
  [geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type= EDGE2
  []
[]

[Variables]
  [T]
    initial_condition = 500
  []
[]

[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
  [th_cond]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = T
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = T
  []
  [heat_source]
     type = NeutronHeatSource
     variable = T
     energy_per_fission = 3.2e-11
     fission_rate = fission_rate
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable= fission_rate
    value = 2e19 # Standard fission_rate fissions/m^3/s
    execute_on = 'initial timestep_begin'
  []
  [th_cond]
    type = MaterialRealAux
    variable = th_cond
    property = thermal_conductivity
    execute_on = 'initial linear'
  []
  [burnup]
    type = BurnupAux
    variable = burnup
    density = 10431.0 # 95% TD
    fission_rate = fission_rate
    execute_on = timestep_begin
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 100.e6'
    y = '500 3000'
  []
[]

[BCs]
   [VariableT]
     type = FunctionDirichletBC
     boundary = 'left right'
     variable = T
     function = temp_ramp
   []
[]

[Materials]
  [fuel_thermal]
    type = MOXThermal
    temperature = T
    burnup = burnup
    thermal_conductivity_model = AMAYA
    oxy_to_metal_ratio = 2.0
    initial_porosity = 0.05
    Pu_content = 0.07
  []
  [density]
    type = ParsedMaterial
    property_name = density
    expression = 10431.0
  []
[]

[Executioner]
  type = Transient
  start_time = 0
  num_steps = 100
  dt = 1.0e6
[]

[Postprocessors]
   [avg_th_cond]
     type = ElementAverageValue
     variable = th_cond
     execute_on = 'initial timestep_end'
   []
   [average_rod_burnup]
     type = ElementAverageValue
     variable = burnup
     execute_on = timestep_end
   []
   [average_fuel_T]
     type = ElementAverageValue
     variable = T
     execute_on = 'initial timestep_end'
   []
 []

[Outputs]
  csv = true
[]

(test/tests/mox_thermal/Amaya/ad_test.i)

# This test case is prepared to test the thermal conductivity using the Amaya MOX model.
#
# The temperature is ramped on all BCs of the unit line from 500 K to 3000K over 100 Ms.
# The fission rate is 2e19 n/m3/s, so that the burnup goes from 0 to
#       ~ 8.6 at.% (82 MWd/kgU) at the end of the simulation.
#
# Thermal conductivity of MOX fuel at 95%TD with a 7% Pu content is computed
# using the Amaya MOX thermal conductivity model.
# The BISON predictions (BISON_k) compared to the expected results (expected_k)
# are summarized (see: post_processing.py --> comparison_results.csv) as:
#
# expected_k      BISON_k      percent_error
# 4.30774878     4.30774878      4.33E-13
# 4.178849627    4.178849627     7.44E-13
# 4.057207008    4.057207008    -4.00E-13
# 3.942475562    3.942475562    -5.55E-13
# 3.834072269    3.834072269     5.77E-13

[GlobalParams]
  order = FIRST
  family = LAGRANGE
[]

[Mesh]
  [geom]
    type = GeneratedMeshGenerator
    dim = 1
    elem_type= EDGE2
  []
[]

[Variables]
  [T]
    initial_condition = 500
  []
[]

[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
  [th_cond]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Kernels]
  [heat]
    type = ADHeatConduction
    variable = T
  []
  [heat_ie]
    type = ADHeatConductionTimeDerivative
    variable = T
  []
  [heat_source]
     type = ADNeutronHeatSource
     variable = T
     energy_per_fission = 3.2e-11
     fission_rate = fission_rate
  []
[]

[AuxKernels]
  [fissionrate]
    type = FissionRateGeneral
    fission_rate_formulation = GENERIC
    variable= fission_rate
    value = 2e19 # Standard fission_rate fissions/m^3/s
    execute_on = 'initial timestep_begin'
  []
  [th_cond]
    type = ADMaterialRealAux
    variable = th_cond
    property = thermal_conductivity
    execute_on = 'initial linear'
  []
  [burnup]
    type = ADBurnupAux
    variable = burnup
    density = 10431.0 # 95% TD
    fission_rate = fission_rate
    execute_on = timestep_begin
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 100.e6'
    y = '500 3000'
  []
[]

[BCs]
   [VariableT]
     type = ADFunctionDirichletBC
     boundary = 'left right'
     variable = T
     function = temp_ramp
   []
[]

[Materials]
  [fuel_thermal]
    type = ADMOXThermal
    temperature = T
    burnup = burnup
    thermal_conductivity_model = AMAYA
    oxy_to_metal_ratio = 2.0
    initial_porosity = 0.05
    Pu_content = 0.07
  []
  [density]
    type = ADParsedMaterial
    property_name = density
    expression = 10431.0
  []
[]

[Executioner]
  type = Transient
  start_time = 0
  num_steps = 100
  dt = 1.0e6
[]

[Postprocessors]
   [avg_th_cond]
     type = ElementAverageValue
     variable = th_cond
     execute_on = 'initial timestep_end'
   []
   [average_rod_burnup]
     type = ElementAverageValue
     variable = burnup
     execute_on = timestep_end
   []
   [average_fuel_T]
     type = ElementAverageValue
     variable = T
     execute_on = 'initial timestep_end'
   []
 []

[Outputs]
  csv = true
[]

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