# Thermal Properties for UPuZr Fuel

Material that calculates the thermal conductivity and specific heat for U-Pu-Zr fuels based on mole fractions, porosity, and temperature.

## Description

The material model ThermalUPuZr calculates the thermal conductivity and heat capacity of U-Pu-Zr, where and refers to the weight fraction of Pu and Zr respectively. The material model accepts either constant values for the weight fractions and , or can be coupled to molar fractions that in turn can be used by other kernels or material models. If atom fractions are provided, ThermalUPuZr converts them into weight fractions: (1) where is the atomic weight of each element in kg/mol, and is the average atomic mass of the fuel mixture in kg/mol.

There are several different models available for calculation of thermal conductivity and heat capacity for unirradiated fuel as described in the following sections.

## Thermal Conductivity Models

### Billone Thermal Conductivity Model

A generic model for the thermal conductivity of U, U-Zr and U-Pu-Zr alloys is given by Billone et al. (1968). The thermal conductivity of unirradiated fuel in units of W/m-K is given by (2) where is the temperature in Kelvin and , , and are temperature coefficients. These coefficients are given by (3) (4) (5) where and are the weight fractions of plutonium and zirconium respectively in the fuel mixture.

Sodium infiltration in to the fuel is modeled per Karahan (2009) as follows (6) where T is in Celsius and is in W/m/K. The porosity of the fuel is corrected to account for pores filled by the sodium coolant with (7) where is the fuel porosity and is the fraction of the fuel which is filled with sodium. The corrected thermal conducitivity that accounts for sodium infiltration is (8)

Sodium infiltration only exists for the Billone model currently.

### Galloway thermal conductivity model

A model for the thermal conductivity of U-Pu-Zr fuel with any concentration of constituents is given by Galloway et al. (2015). Data used to develop the empirical model for thermal conductivity of U-Pu-Zr and U-Zr fresh fuels are obtained from Janney and Papesch (2015), Kim and Hofman (2003), Touloukian et al. (1971), Takahashi et al. (1988), and Bauer (1995). The basis for the model is derived from the formulation given by Touloukian et al. (1971) with coefficient adjustments to minimize the standard deviation of error between data and the current empirical model. The model consists of calculation of the thermal conductivity of each constituent, the thermal conductivity of the binaries U-Zr and Pu-Zr, and finally the thermal conductivity of the ternary UPuZr.

The thermal conductivities in W/m-K for each constituent is calculated by (9)

(10)

(11) where is temperature in K. The binary thermal conductivities are calculated by (12) where the adjusted weight fractions for the binary formulations are given as (13)

The correction terms for the binaries are given as (14)

Finally, the ternary thermal conductivity is calculated by (15)

This empirical model gives an average error of -0.02 W/m-K with a standard deviation of 1.46 W/m-K.

### Ternary thermal conductivity model: Kim

The Ternary thermal conductivity model is very similar to the model used by Galloway, Eq. 15, albeit with different coefficients and different formulation of the ternary thermal conductivity. The original formulation comes from Kim et al. (2014).

The thermal conductivities in W/m-K for each constituent is calculated using Eq. 9 and Eq. 11, with the formulation for plutonium as (16) where is temperature in K. The U-Zr binary thermal conductivity is calculated by: (17) where the adjusted weight fractions for the binary formulations are given as: (18) The correction terms for the U-Zr binary are given as, (19)

The thermal conductivity of the ternary fuel is then (20) where is the weight fraction of plutonium in the fuel and the plutonium thermal conductivity correction is given by (21)

### Ternary thermal conductivity model: LANL

Recent work has been applied to extend Kim's model to more data, resulting in new coefficients (Matthews and Unal, 2015). These coefficients are also available as a separate model, designated LANL: (22)

Although only slightly different, when the updated coefficients are plugged into the remainder of the ternary model, the calculation of thermal conductivity results in a standard deviation of less than 1 W/m-K.

### Porosity correction

When taking into account irradiation effects through the introduction and growth of porosity, the thermal conductivity becomes, (23) where (24) The porosity is calculated by UPuZrVolumetricSwellingEigenstrain (or the Solid Mechanics version, VSwellingUPuZr) and passed to ThermalUPuZr as an auxkernel. The porosity constant is typically taken as 2.5 for conservatism as recommended by Billone et al. (1968).

## Specific Heat Capacity Models

### Karahan Heat Capacity

The specific heat capacity of U-Pu-Zr alloys are dependent upon the phase (, or ) as per Karahan (2009) where the transition temperatures are taken from Savage (2006), as T = 600 C and T = 650 C.

For the phase: (25) For the phase: (26) For the transition phase, interpolation has been performed such that, (27) In the above equations for specific heat capacity T is the temperature in Celsius and is the average atomic mass of the fuel mixture in kg.

It is worth noting that this model is presented by Karahan (2009) but fails to divide the leading constant term by , leading to miscalculation of the original data from Savage (2006).

### Savage Heat Capacity

The correlation for heat capacity from Savage is split into the low temperature region, and the high temperature region: (28)

(29) where is temperature in C, and is given in cal/mol-C. A simple linear interpolation is used for the region: (30) where and are the transition temperature 600 C and 650 C respectively.

Lastly, is converted from cal/(mol-C) to J/(kg-K) by multipling Eq. 28, Eq. 29, or Eq. 30 by 4.184 (J/cal) and 0.205 (kg/mol). It is important to note that the kg to mol conversion is kept constant, as it applies directly to the data captured by Savage. Changing the conversion factor will result in a dependency on the concentrations of Pu and Zr in the fuel that is not based on data, as is the case with the Karahan model described above.

## Example Input Syntax

[./thermalUPuZr]
type = ThermalUPuZr
block = 0
temp = temp
X_Pu = 0.16
X_Zr = X_Zr
porosity = porosity
outputs = all
thcond_model = billone
spheat_model = karahan
output_properties = 'thermal_conductivity thermal_conductivity_dT specific_heat'
[../]
(test/tests/thermalTests/thermalUPuZr/billone_karahan.i)

## Input Parameters

• tempCoupled temperature.

C++ Type:std::vector

Description:Coupled temperature.

### Required Parameters

Default:0.0041

C++ Type:double

• porosity_factor2.5Factor used when calculating porosity correction (1-P)/(1-factor*P).

Default:2.5

C++ Type:double

Description:Factor used when calculating porosity correction (1-P)/(1-factor*P).

• A_U0.238029Atomic weight of uranium [kg/mol].

Default:0.238029

C++ Type:double

Description:Atomic weight of uranium [kg/mol].

• computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

Default:True

C++ Type:bool

Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

• zr_weightConstant weight fraction of zirconium.

C++ Type:double

Description:Constant weight fraction of zirconium.

• spheat_modelkarahanSpecific heat model. Choices are: karahan savage

Default:karahan

C++ Type:MooseEnum

Description:Specific heat model. Choices are: karahan savage

• X_ZrCoupled zirconium molar fraction.

C++ Type:std::vector

Description:Coupled zirconium molar fraction.

• X_PuCoupled plutonium molar fraction.

C++ Type:std::vector

Description:Coupled plutonium molar fraction.

• initiating_porosity0.24Porosity at which sodium infiltration begins.

Default:0.24

C++ Type:double

Description:Porosity at which sodium infiltration begins.

• thcond_modelbilloneThermal conductivity model. Choices are: billone galloway lanl kim

Default:billone

C++ Type:MooseEnum

Description:Thermal conductivity model. Choices are: billone galloway lanl kim

• k_scalar1Scalar multiplied against the calculated thermal conductivity.

Default:1

C++ Type:double

Description:Scalar multiplied against the calculated thermal conductivity.

• pu_weightConstant weight fraction of plutonium.

C++ Type:double

Description:Constant weight fraction of plutonium.

• Na_depth0Pellet depth sodium has infiltrated as a percentage of fuel radius.

Default:0

C++ Type:double

Description:Pellet depth sodium has infiltrated as a percentage of fuel radius.

• boundaryThe list of boundary IDs from the mesh where this boundary condition applies

C++ Type:std::vector

Description:The list of boundary IDs from the mesh where this boundary condition applies

• porosity0Porosity material property name

Default:0

C++ Type:MaterialPropertyName

Description:Porosity material property name

• A_Zr0.091224Atomic weight of zirconium [kg/mol].

Default:0.091224

C++ Type:double

Description:Atomic weight of zirconium [kg/mol].

• A_Pu0.244Atomic weight of plutonium [kg/mol].

Default:0.244

C++ Type:double

Description:Atomic weight of plutonium [kg/mol].

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

• 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

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.

• control_tagsAdds user-defined labels for accessing object parameters via control logic.

C++ Type:std::vector

Description:Adds user-defined labels for accessing object parameters via control logic.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

• implicitTrueDetermines whether this object is calculated using an implicit or explicit form

Default:True

C++ Type:bool

Description:Determines whether this object is calculated using an implicit or explicit form

• 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 computeSubdomainProperties() 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

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 computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

• output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)

C++ Type:std::vector

Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

• outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object

Default:none

C++ Type:std::vector

Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

## References

1. T H Bauer. In-Pile Measurement of the Thermal Conductivity of Irradiated Metallic Fuel. Nuclear Technology, 110(3):1â€“15, 1995.[BibTeX]
2. M. C. Billone, Y. Y. Liu, E. E. Gruber, T. H. Hughes, and J. M. Kramer. Status of Fuel Element Modeling Codes for Metallic Fuels. In Proceedings American Nuclear Society Internaionl Conference on Reliable Fuels for Liquid Metal Reactors. Tucson, Arizona, September 7â€“11 1968.[BibTeX]
3. J Galloway, C Unal, N Carlson, D Porter, and S Hayes. Modeling constituent redistribution in Uâ€“Puâ€“Zr metallic fuel using the advanced fuel performance code BISON. Nuclear Engineering and Design, 286:1â€“17, May 2015.[BibTeX]
4. Dawn E Janney and Cynthia A Papesch. FCRD Transmutation Fuels Handbook 2015. Technical Report INL/EXT-15-36520, Idaho National Laboratory, September 2015.[BibTeX]
5. Aydin Karahan. Modeling of thermo-mechanical and irradiation behavior of metallic and oxide fuels for sodium fast reactors. PhD thesis, Massachusetts Institute of Technology, June 2009.[BibTeX]
6. Yeon Soo Kim, Tae Won Cho, and Dong-Seong Sohn. Thermal conductivities of actinides (U, Pu, Np, Cm, Am) and uranium-alloys (U-Zr, U-Pu-Zr and U-Pu-TRU-Zr). Journal of Nuclear Materials, 445(1-3):272â€“280, February 2014.[BibTeX]
7. Yeon Soo Kim and G. L. Hofman. AAA Fuels Handbook. Technical Report, Argonne National Laboratory, 2003.[BibTeX]
8. C Matthews and C Unal. Unpublished work. 2015.[BibTeX]
9. H. Savage. The heat content and specific heat of some metallic fast-reacdtor fuels containing plutonium. Journal of Nuclear Materials, 25:583â€“594, 2006. doi:10.1016/0022-3115(68)90168-2.[BibTeX]
10. Yoichi Takahashi, Michio Yamawaki, and Kazutaka Yamamoto. Thermophysical properties of uranium-zirconium alloys. Journal of Nuclear Materials, 154(1):141â€“144, June 1988.[BibTeX]
11. Y S Touloukian, R W Powell, C Y Ho, and P G Klemens. \textbf Thermophysical Properties of Matter - The TPRC Data Series. Volume 2. Thermal Conductivity - Nonmetallic Solids. Defense Technical Information Center, New York, NY, January 1971.[BibTeX]